Research Article Model Building and Optimization Analysis of MDF Continuous …downloads.hindawi.com/journals/mpe/2016/1327235.pdf · 2019. 7. 30. · for solving nonsmooth convex

Research ArticleModel Building and Optimization Analysis of MDF ContinuousHot-Pressing Process by Neural Network

Qingfa Li1 Yaqiu Liu1 and Liangkuan Zhu2

1School of Information and Computer Engineering Northeast Forestry University Harbin 150040 China2College of Electromechanical Engineering Northeast Forestry University Harbin China

Correspondence should be addressed to Yaqiu Liu yaqiuliugmailcom

Received 8 March 2016 Accepted 9 August 2016

Academic Editor Yakov Strelniker

Copyright copy 2016 Qingfa Li et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a one-layer neural network for solving a class of constrained optimization problems which is brought forward fromthe MDF continuous hot-pressing process The objective function of the optimization problem is the sum of a nonsmooth convexfunction and a smooth nonconvex pseudoconvex function and the feasible set consists of two parts one is a closed convex subsetof 119877119899 and the other is defined by a class of smooth convex functions By the theories of smoothing techniques projection penaltyfunction and regularization term the proposed network is modeled by a differential equation which can be implemented easilyWithout any other condition we prove the global existence of the solutions of the proposed neural network with any initial point inthe closed convex subsetWe show that any accumulation point of the solutions of the proposed neural network is not only a feasiblepoint but also an optimal solution of the considered optimization problem though the objective function is not convex Numericalexperiments on the MDF hot-pressing process including the model building and parameter optimization are tested based on thereal data set which indicate the good performance of the proposed neural network in applications

1 Introduction

Medium density fibreboard (MDF) finds many applicationsin wood industries because of its favorable properties suchas surface characteristics dimensional stability and excel-lent machinability [1 2] In the MDF hot-pressing processmany physical processes are involved and the complexity ofthis operation arises from the fact that they are coupledHot-pressing process is one of the key procedures in theproduction of MDF which influences the utilization ratioof energy and resource With the decreased resource oftimber and the increased demand of MDF it is of greatimportant to analyze the experimental data effectively andreasonably find the main factors among the many indexesof MDF and establish the relation models on the propertiesof slab the parameters in the hot-pressing process and themain indexes of MDF These relation models not only canhelp the staff give reasonable prediction and a reliabilityassessment to the hot-pressing process according to the actualprocess parameters but also provide a theoretical basis forthe setting and adjusting of the main factors in hot-pressing

process according to the actual demand of MDF propertiesSo optimization models and methods have been importanttools for the optimization control and scheduling of the hot-pressing of plates

Real-time online solutions of optimization problems aredesired in many engineering and scientific applications Onepossible and very promising approach to solve the real-timeoptimization problems is to apply artificial neural networks[3ndash5] With the resemblance brains neural networks canbe implemented online by hardware and have become animportant technical toll for solving optimization problemsfor example [3 4 6ndash9] Based on the gradient method theHopfield neural networks proposed in [4 5] are the twoclassical recurrent neural networks for linear and nonlin-ear programming whereafter in addition to the gradientmethod many types of neural networks are designed suchas the Lagrangian neural networks [10] the projection-type neural networks [11 12] the dual network [13] andthe stochastic neural network [14] Projection method isan effective and simple method for solving the constraintsHowever it is impossible to solve the general constraints by

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1327235 16 pageshttpdxdoiorg10115520161327235

2 Mathematical Problems in Engineering

projection method Then Lagrangian and penalty methodsare introduced into networks Based on the Lagrangian func-tionmethod Lagrangian networks were proposed for solvingthe optimization problems [8 10] with general constraintsBut the Lagrangian network increases the dimension of thenetworks along with the number of the constraints In recentyears recurrent neural networks based on penalty methodwere widely investigated for solving optimization problemsThe neural networks for smooth optimization problems cannot solve nonsmooth optimization problems because thegradients of the objective and constrained functions arerequired in such neural networks The generalized nonlinearprogramming circuit (G-NPC) in [15] can be considered asa natural extension of nonlinear programming circuit (NPC)for solving nonsmooth convex optimization problems withinequality constraints But the nonempty interior of feasibleregion and large enough penalty parameters are needed forthe network in [15] In order to overcome the nonemptyassumption of the interior of feasible region Bian and Xue[6] proposed a recurrent neural network for nonsmoothconvex optimization based on penalty function methodThe efficiency of the neural networks for solving convexoptimization problems relies on the convexity of functionsA neural network for nonconvex quadratic optimization ispresented in [16] Some neural networks modeled by differ-ential inclusion were also proposed for some nonsmooth andnonconvex optimization problems [6 17] To overcome thedifferential inclusion smoothing techniques are introducedinto the neural networks The main feature of smoothingmethod is to approximate the nonsmooth functions by a classof smooth functions Thus the neural network constructedby the smoothing techniques is modeled by a differentialequation which can be implemented easily in circuits andmathematical software [18]

In this paper we propose a neural network model forsolving the optimization problem brought forward from theMDF continuous hot-pressing automatic control system InSection 2 some notations and necessary preliminary resultsare listed In Section 3 based on the SVM theory withthe existing linear and nonlinear kernel functions we givean optimization problem which includes the problems forbuilding the models of MDF continuous hot-pressing systemand optimizing the MDF performance indexes as specialcases In order to build up the relation models on theproperties of the slab technical parameters in hot-pressingprocess and the performance indexes of MDF when the ker-nel function is positive definite or semipositive definite thecorresponding optimization problem is a constrained convexproblem otherwise it is a nonconvex problemThe optimiza-tion problem for optimizing the performance parametersis a nonconvex constrained optimization problem but itsobjective function is pseudoconvex due to the appropriatechoice of kernel functions In Section 4 we propose a neuralnetwork based on the penalty function method projectionmethod and smoothing techniques The proposed networkis modeled by a nonautomatic differential equation ByLyapunovmethod we prove that the solution of the proposednetwork is global existent and convergent to the feasible setof the considered optimization problems Moreover due to

the pseudoconvexity of the objective function and the con-vexity of the constraint the proposed network also convergesto the optimal solution set of the optimization problem InSection 5 based on the existing data set we use the proposednetwork into the model building and parameter optimizingproblems of hot-pressing system which validates the goodperformance of the obtained results in this paper

Notations 119877+= [0 +infin) Given column vectors 119909 = (119909

1

1199092 119909

119899)119879 and 119910 = (119910

1 119910

2 119910

119899)119879 ⟨119909 119910⟩ = 119909

119879

119910 =

sum119899

119894=1119909119894119910119894is the scalar product of 119909 and 119910 119909

119894denotes the

119894th element of 119909 119909 denotes the Euclidean 2-norm definedby 119909 = (sum

119899

119894=11199092

119894)12 For a closed convex subset Γ sube 119877

119899dist(119909 Γ) is the distance from 119909 to Γ defined by dist(119909 Γ) =min

119910isinΓ119909 minus 119910

2 Preliminaries

In this section we state some definitions and propertiesneeded in this paper We refer the readers to [19ndash21]

21 Support Vector Regression Kernels were regarded as afunction with the formulation of inner product and havebeen a powerful tool in machine learning for their superiorperformance over a wide range of learning problems suchas isolated handwritten digit recognition text categorizationand face detection [21 22]

LetX be a nonempty set and 119870 X timesX rarr 119877 be a real-valued and symmetric function With the kernel matrix K =

(119896(119909119894 119909

119895))119899

119894119895=1 119870 is said to be a positive semidefinite kernel

ifK is positive semidefinite for any 119899 isin N and 1199091 119909

2 119909

119899isin

XWe call119870 an indefinite kernel if there exist1199091 119909

2 119909

119899isin

X and V 119908 isin 119877119899 such that V119879KV lt 0 and 119908119879K119908 gt 0In what follows we list some widely used kernels

(i) Gaussian radial basis function kernel 119870(119909 119910) =

exp(minus119909 minus 119910221205902)

(ii) Polynomial kernel 119870(119909 119910) = (⟨119909 119910⟩ + 119888)119901 119888 ge 0

119901 ge 1

(iii) Sigmoid kernel 119870(119909 119910) = tanh(120581⟨119909 119910⟩ + V) 120581 gt 0V lt 0

22 Smoothing Approximation Smoothing approximation isan effective method for solving nonsmooth optimizationproblems and has been widely used in the past decades Themain feature of smoothing method is to approximate thenonsmooth functions by a class of parameterized smoothfunctions In this paper we adopt the smoothing functiondefined as follows

Definition 1 (see [23]) Let 119892 119877119899

rarr 119877 be a continuousfunction One calls 119877119899 times119877

+rarr 119877 a smoothing function of

119892 if (sdot 120583) is continuously differentiable for any fixed 120583 gt 0and lim

119911rarr119909120583darr0(119911 120583) = 119892(119909) holds for any 119909 isin 119877119899

Mathematical Problems in Engineering 3

Chen and Mangasarian constructed a class of smoothapproximations of the function (119904)

+fl max0 119904 by convo-

lution [20 24] as follows Let 120588 119877 rarr 119877+be a piecewise

continuous density function satisfying

120588 (119905) = 120588 (minus119905)

120581 fl int

infin

minusinfin

|119905| 120588 (119905) 119889119905 lt infin

(1)

Then

120601 (119905 120583) fl int

infin

minusinfin

(119904 minus 120583119905)+120588 (119905) 119889119905 (2)

from 119877 times 119877+to 119877

+is well defined

By different density functions many popular smoothingfunctions of (119904)

+can be derived such as

1206011(119904 120583) = 119904 + 120583 ln (1 + 119890minus119904120583)

1206012(119904 120583) =

1

2(119904 + radic1199042 + 41205832)

1206013(119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

1206014(119904 120583) =

119904 +120583

2119890minus119904120583 if 119904 gt 0

120583

2119890minus119904120583 if 119904 le 0

(3)

where 1206011is the neural networks smoothing function 120601

2is

called the CHKS (Chen-Harker-Kanzow-Smale) smoothingfunction120601

3is called the uniform smoothing function and120601

4

is called the Picard smoothing function The four functionsbelong to the class of the Chen-Mangasarian smoothingfunctions

Many nonsmooth functions can be reformulated by usingthe plus function We list some of them as follows

|119904| = (119904)++ (minus119904)

+

max (119904 119905) = 119904 + (119905 minus 119904)+

min (119904 119905) = 119904 minus (119904 minus 119905)+

mid (119904 ℓ 119906) = min (max (ℓ 119904) 119906) for given ℓ 119906

(4)

So we can define a smoothing function for the abovenonsmooth functions by a smoothing function of (119904)

+

From Theorem 961 and Corollary 847(b) in [20] when119892 119877

119899

rarr 119877 is locally Lipschitz continuous at 119909 thesubdifferential associated with a smoothing function

119866(119909) = con V | nabla

119909 (119909

119896

120583119896) 997888rarr V for 119909119896 997888rarr 119909 120583

119896

darr 0

(5)

is nonempty and bounded and 120597119892(119909) sube 119866(119909) where ldquoconrdquo

denotes the convex hull In [20 23] it is shown that manysmoothing functions satisfy the gradient consistency

120597119892 (119909) = 119866(119909) (6)

which is an important property of the smoothing methodsand guarantees the convergence of smoothing methods withadaptive updating schemes of smoothing parameters to astationary point of the original problem

23 Pseudoconvex Function Pseudoconvex function is a classof functions which may be nonsmooth or nonconvex butbrings us the opportunity to find the optimal solutions

Definition 2 (see [25]) Let X be a nonempty convex subsetof 119877119899 A function 120601 is said to be pseudoconvex on X if forany 119909 119910 isin X one has

exist120585 (119909) isin 120597120601 (119909)

st ⟨120585 (119909) 119910 minus 119909⟩ ge 0 997904rArr

120601 (119910) ge 120601 (119909)

(7)

Many nonconvex functions in application are pseudocon-vex such as the Butterworth filter function fraction functionand density functionOf particular interest in this paper is thefact that the Gaussian function

ℎ (119909) = minus exp(minus119899

sum

119894=1

1199092

119894

1205902

119894

) (8)

with 120590119894gt 0 119894 = 1 2 119899 is pseudoconvex on 119877119899

24 Project Operator LetX be a closed convex subset of 119877119899Then the projection operator toX at 119909 is defined by

119875X (119909) = argmin119906isinX

119906 minus 1199092 (9)

and satisfies the following inequalities

⟨V minus 119875X (V) 119875X (V) minus 119906⟩ ge 0 forallV isin 119877119899 119906 isin X

1003817100381710038171003817119875X (119906) minus 119875X (V)1003817100381710038171003817 le 119906 minus V forall119906 V isin 119877119899

(10)

(i) Suppose Ω = 119909 isin 119877119899 119906 le 119909 le V with 119906 V isin 119877119899 cup plusmninfin

then 119875Ω(119909) = (119901

1 119901

2 119901

119899)119879 can be expressed by

119901119894=

119906119894119909119894lt 119906

119894

119909119894119906119894le 119909

119894le V

119894

V119894

V119894lt 119909

119894

(11)

(ii) Suppose Ω = 119909 119860119909 = 119887 with 119860 isin 119877119903times119899 of full row rank

and 119887 isin 119877119903 then 119875Ω(119909) = 119909 minus 119860

119879

(119860119860119879

)minus1

(119860119909 minus 119887)

3 Optimization Problems in MDF ContinuousHot-Pressing Process

In this section we will give the optimization model consid-ered in this paper First by the optimization and supportvector machine theories we show the optimization modelsfor building up the relationships in MDF continuous hot-pressing process Then another optimization model for


optimizing the parameters in the MDF continuous hot-pressing process is obtained Thus we express these twokinds of problems into a uniform formulation which is theoptimization problem considered in Section 3

31 Optimization Problem for Building up the Models of MDFContinuousHot-Pressing Process DenoteX sub 119877

119901 andY sub 119877

as two sets where 119909119894 = (1199091198941 119909

119894

2 119909

119894

119901) isin X is the attribute

vector on behalf of the hot-pressing plate properties 119910119894 isin Yindicates the values of the qualities of hot-pressing plate Let119878 = (119904

1

1199042

119904119899

) be the training data set of hot-pressingprocess where 119904119894 = (119909119894 119910119894) obeys the unknown distributionand is IID (independent and identically distributed) Basedon the support vector machine theory and the training dataset we would like to find a nonlinear function 119910 = 119891(119909) suchthat it approximates the training data set as much as possible

With the kernel function 119870 and the Huber loss functionfrom the theory in [21] the dual optimization problem of theprimal optimization problem of SVM is given as

min120572120573

1

2

119899

sum

119894119895=1

(120573119894minus 120572

119894) (120573

119895minus 120572

119895)119870 (119909

119894

119909119895

)

minus

119899

sum

119894=1

(120573119894minus 120572

119894) 119910

119894

+120576

2119862

119899

sum

119894=1

(120573119894minus 120572

119894)2

st119899

sum

119894=1

(120573119894minus 120572

119894) = 0

0 le 120573119894 120572

119894le 119862 119894 = 1 2 119899

(12)

Denote 120572lowast119894and 120573lowast

119894as the optimal solutions of (12) Then the

regression function 119891 can be expressed by

119891 (119909) =

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894)119870 (119909

119894

119909) + 119887 (13)

where 119887 can be calculated by one of the following twomethods

119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895

⟩ + 120576 120572lowast

119895isin (0 119862)

119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895

⟩ minus 120576 120572lowast

119895isin (0 119862)

(14)

Denote 119909119894= 120573

119894minus 120572

119894in (12) then (12) can be reformulated

as

min 1

2119909119879

119876119909 minus 119910119879

119909

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(15)

where 119909 = (1199091 119909

2 119909

119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1

1)119879

isin 119877119899 and 119876 = K+(1205762119862)119868

119899withK = (119870(119909

119894

119909119895

))119899

119894119895=1

For the optimal solution 119909lowast = (119909lowast1 119909

lowast

2 119909

lowast

119899) of (15) we call

119909119894 a support vector if 119909lowast

119894= 0 The optimal solution of (15)

solved by the originalmethods often has almost 100 supportvectors which increases the complexity of the relationmodelslargely Thus we introduce the problem deduced by (15) thatis

min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(16)

with 120574 gt 0 In (43) 120574sum119899

119894=1|119909119894| is often called the regulation

term which is used to control the number of its supportvectors

Define the penalty function

119875 (119909) =

1003817100381710038171003817100381710038171003817

1

119899119890119890119879

119909

1003817100381710038171003817100381710038171003817=radic119899

119899

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=1

119909119894

1003816100381610038161003816100381610038161003816100381610038161003816

(17)

and then 119909 119875(119909) le 0 = 119909 119890119879119909 = 0 From [19 Proposition243] 119909 is an optimal solution of (43) if and only if it is anoptimal solution of the following problem

min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816

st minus 119862119890 le 119909 le 119862119890

(18)

where 120590 = 119862119876 + 119910radic119899 + 120574 + 1 Thus we can build up therelation models of hot-pressing process by solving problem(18)

32 Optimization Problem for Optimizing the Parameters inMDF Hot-Pressing Process Based on the relationships builtup in Section 31 we focus on themodulus of rupture (MOR)modulus of elasticity (MOE) and internal bonding strength(IBS) of hot-pressing plate by optimizing the process param-eters and slab attributes Suppose the regression functions ofMORMOS and IBSwith respect to some relative parametersbased on the Gaussian radial basis function kernel are

1198911(119909) fl

119899

sum

119894=1

120588lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

3(IBS)

(19)

and based on the linear polynomial kernel are

1198911(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894⟨119911

119894 119909⟩ + 119887

3(IBS)

(20)


where 1199111 119911

2 119911

119899are the variables in the data set for

regression and 119909 = (1199091 119909

2 119909

3 119909

4)119879 indicates the indepen-

dent variable with hot-pressing temperature 1199091 hot-pressing

pressure 1199092 hot-pressing time 119909

3 and moisture content 119909

4

The regression functions in (19) satisfy the following twoproperties

(i) 119891119894is continuously differentiable on 119877119899 119894 = 1 2 3

(ii) 119891119894and minus119891

119894are not convex but minus119891

119894is pseudoconvex on

119877119899 119894 = 1 2 3

And the regression functions in (20) satisfy the following twoproperties


(ii) 119891119894and minus119891

119894are convex on 119877119899 119894 = 1 2 3

From the physical significance of MOR MOS and IBSwe suppose that the larger the numbers of MOR MOS andIBS the better the quality of hot-pressing plate In order toadopt the different demand on the indexes of the hot-pressingplate in different applications we consider the following twocases in this part

First we focus onmaximizing a single performance indexof the hot-pressing plate when the other two performanceindexes are within the certain areas If we want to optimizethe IBS the corresponding optimization model for this casecan be expressed by

min minus 1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le 119891

1(119909) minus 119910

lowast

1le 120576

1

minus 1205802le 119891

2(119909) minus 119910

lowast

2le 120576

2

(21)

where ]1 ]2 ]3 ]4gt 0 indicate the upper bounds of hot-

pressing temperature hot-pressing pressure hot-pressingtime andmoisture content and [119910lowast

1minus120580

1 119910

lowast

1+120576

1] [119910lowast

2minus120580

2 119910

lowast

2+

1205762] are the feasible regions of MOR and MOS respectivelyIn order to let problem (21) be solved effectively we

let the objective function 1198913be with the Gaussian radial

basis function and the regression functions 1198911and 119891

2in the

constraints are with the linear polynomial kernel Then (21)is

min minus

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1le 120576

1

minus 1205802le

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2le 120576

2

(22)

which is a pseudoconvex optimization problem with convexconstraints

Second we would like to optimize the MOR MOS andIBS synthetically For this demand we consider the followingoptimization model

min minus1205821

119891lowast

1

1198911(119909) minus

1205822

119891lowast

2

1198912(119909) minus

1205823

119891lowast

3

1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

(23)

where 1205821 120582

2 120582

3gt 0 indicate the importance of MOR MOS

and IBS ]1 ]2 ]3 ]4are with the same meaning as in (21)

and 119891lowast1 119891lowast

2 and 119891lowast

3are the expected values of MOR MOS

and IBS In particular if MOR MOS and IBS are with thesame importance in the quality of the hot-pressing plate wecan let 120582

1= 120582

2= 120582

3= 13 and we can let 120582

1= 12

1205822= 13 and 120582

3= 16 if the importance of MOR MOS

and IBS is strictly monotone decreasing Similar to the kernelfunctions in problem (21) we let119891

1119891

2 and119891

3in problem (23)

be with the Gaussian radial basis function which means thatproblem (23) is also a pseudoconvex optimization problemwith convex constraints

33 General Model Based on analysis in Sections 31 and32 we consider the following minimization problem in thispaper

minimize 119891 (119909) fl Θ (119909) + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816

subject to 119909 isin Ω fl 119909 119886 le 119909 le 119887

119892119894(119909) le 0 119894 = 1 119898

(24)


where 120574 120590 ge 0 119886 119887 isin 119877119899 with 119886 lt 119887 Θ 119877

119899

rarr 119877

is continuously differentiable and pseudoconvex on 119877119899 and119892119894 119877

119899

rarr 119877 (119894 = 1 2 119898) is continuously differentiableand convex on 119877119899

On the one hand when Θ(119909) = (12)119909119879119876119909 minus 119910119879119909 119886 flminus119862119890 119887 fl 119862119890 and 120574 and 120590 are defined as in (18) then problem(24) without 119892

119894reduces to problem (18) On the other hand

if we let

Θ (119909) fl minus

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

119886 fl 0

119887 fl (]1 ]2 ]3 ]4)119879

(25)

1198921(119909) fl minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

(26)

1198923(119909) fl minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(27)

then problem (24) reduces to problem (22) Similar reformu-lation can be done for problem (23) by (24)

Therefore problem (24) considered in this paper includesthe optimizationmodels for building up the relationships andoptimizing the relative parameters in MDF continuous hot-pressing process

Inwhat follows we denoteF as the feasible region of (24)that is

F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)

andM is the optimal solution set of (24)

4 Main Results

41 Proposed Neural Network In this subsection we proposea one-layer recurrent neural network for solving problem(24) where we combine the penalty function and projectionmethods to solve the constraints and use the smoothingtechniques to overcome the nonsmoothness of the objectivefunction and penalty function


119901 (119909) =

119898

sum

119894=1

max 0 119892119894(119909) (29)

Then 119909 isin Ω 119901(119909) le 0 = FFrom the smoothing functions in (25) for the plus

function we define the smoothing function of 119901 as

(119909 120583) =

119898

sum

119894=1

120601 (119892119894(119909) 120583) (30)

where

120601 (119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

(31)

Form the results in [23] 120601(119904 120583) owns the followingproperties

Lemma 3 (see [23]) (i) For any 119904 isin 119877 120601(119904 sdot) is continuouslydifferentiable and 120601(sdot 120583) is also continuously differentiable forany fixed 120583 gt 0

(ii) 0 le nabla120583120601(119904 120583) le 1 forall119904 isin 119877 forall120583 isin (0 +infin)

(iii) 0 le 120601(119904 120583) minus |119904| le 1205834 forall119904 isin 119877 forall120583 isin (0 +infin)(iv) 120601(sdot 120583) is convex for any fixed 120583 gt 0 and

lim119904rarr119905120583darr0

nabla119904120601(119904 120583) sube 120597max0 119905

Then has the following properties

Lemma 4 is a smoothing function 119901 and satisfies thefollowing

(i) (sdot 120583) is convex for any fixed 120583 gt 0(ii) lim

119911rarr119909120583darr0nabla119911(119911 120583) sube 120597119901(119909)

(iii) 0 le nabla120583(119909 120583) le 119898 forall119909 isin 119877119899 120583 isin (0 +infin)

Next by the smoothing function for the absolute valuefunction | sdot |

120579 (119904 120583) =

|119904| if |119904| ge 120583

1199042

2120583+120583

2if |119904| lt 120583

(32)

we define

(119909 120583) fl Θ (119909) + 120574

119899

sum

119894=1

120579 (119909119894 120583) + 120590120579 (119890

119879

119909 120583) (33)

Since 120579(119904 120583) = 120601(119904 120583) + 120601(minus119904 120583) 120579(119904 120583) owns allproperties in Lemma 8 and the following results hold

Lemma 5 is a smoothing function 119891 in (24) with thefollowing properties

(i) (sdot 120583) is pseudoconvex for any fixed 120583 gt 0

(ii) lim119911rarr119909120583darr0

nabla119909(119909 120583) sube 120597119891(119909)

(iii) 0 le nabla120583(119909 120583) le 120574119899 + 120590 forall119909 isin 119877119899 120583 isin (0 +infin)

From the projected gradient method and the viscosityregularization method we introduce the following neuralnetwork to solve (24)

(119905) isin minus119909 (119905) + 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(34)

where 120583(119905) = 119890minus119905 and 120576(119905) = 1(119905 + 1) with 1205760gt 0


120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla

Figure 1 Simple block structure of proposed network (37)

By the definitions for (34) can be expressed as

(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)

To implement (34) by circuits we can use the reformu-lated form of (34) as follows


119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)

Equation (34) can be seen as a network with three input andthree output variables that are 119909(119905) 120583(119905) and 120576(119905) A simpleblock structure of the proposed network (37) implementedby circuits is presented in Figure 1

42 Theoretical Analysis In this subsection we study somenecessary dynamical and optimality properties of proposednetwork (34) for solving (24)

The global existence of the solutions of (34) is a necessarycondition for its usability in optimization With an initial

point 1199090isin Ω the solution of (34) is global existentMoreover

the uniqueness of the solution of (34) with 1199090isin Ω is proved

under some conditions The proposed network (34) can beimplemented in circuits and mathematical software Thenthe feasibility and optimality of network (34) for optimizationproblem (24) are proved theoretically

We call 119909 [0 119879) with T gt 0 a solution of (34) if 119909 isabsolutely continuous on [0 119879) and satisfies (34) everywhere

Proposition 6 For any initial point 1199090isin Ω there is a global

solution of (34) defined on [0 +infin) and it satisfies

119909 (119905) isin Ω forall119905 isin [0 +infin) (38)

Proof Since the right hand function in network (34) iscontinuous about 119909 and 119905 there are 119879 gt 0 and an absolutecontinuous function 119909 [0 119879) rarr 119877

119899 such that 119909(119905) satisfies(34) for all 119905 isin [0 119879)

Denote 120579(119905) = 119875Ω[119909(119905) minus nabla

119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)

120583(119905))] Then (34) can be rewritten as

(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)

A simple integration procedure of the above equation gives

119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)

which can be rewritten as

119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905

minus 1)119889119904 = 1 120579(119904) isin Ω forall0 le 119904 le 119905 and 1199090isin Ω we

obtain that 119909(119905) isin Ω forall119905 isin [0 119879)By the boundedness of Ω and the extension theory this

solution of (34) can be extended Thus the solution of (34)with initial point 119909

0isin Ω is globally existent Similarly we can

obtain that


Some Lipschitz condition is often used to guaranteethe uniqueness of the solution of a neural network Inwhat follows we give a sufficient condition to ensure theuniqueness of the solution of (34) with initial point 119909

0isin Ω

Proposition 7 For any initial point 1199090isin Ω if nabla

119909(sdot 120583) and

nabla119909(sdot 120583) are locally Lipschitz continuous for any fixed 120583 isin

(0 1] then the solution of neural network (34) is unique

Proof By Proposition 6 the solutions of (34) with initialpoint119909

0isin Ω exist globally and satisfy119909(119905) isin Ωforall119905 isin [0 +infin)

Suppose that there exist two solutions 119909 [0 +infin) rarr 119877119899

and 119910 [0 +infin) rarr 119877119899 of (34) with initial point 119909

0isin Ω

and suppose there exists such that = inf119905ge0119909(119905) =119910(119905)

119905 By theboundedness of Ω there is 119877 gt 0 such that 119909(119905) le 119877 and119910(119905) le 119877 forall119905 isin [0 +infin)


Then there is 119871 gt 0 such that10038161003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla

119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

1003816100381610038161003816nabla119909 (119909 (119905) 120583 (119905)) minus nabla119909 (119910 (119905) 120583 (119905))1003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)

Differentiating (12)119909(119905)minus119910(119905)2 along the two solutionsof (34) by the Lipschitz continuity of 119875

Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)

Applying Gronwallrsquos inequality into the integration of theabove inequality it gives 119909(119905) = 119910(119905) forall119905 isin [119905 + 1] whichleads to a contradiction Therefore the solution of (34) withinitial point 119909

0isin Ω is unique

Lyapunov method is employed to analyze the perfor-mance of (34) Here we introduce the following two Lya-punov energy functions

119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)

The above two Lyapunov functions satisfy the followingestimations along the solutions of (34)

Lemma 8 (i) The derivative of 119864(119909 119905) along the solution of(34) can be calculated by

119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)

(ii) The derivative of 119866(119909 119905) along the solution of (34) can becalculated by

119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)

Proof (i) Differentiating119864(119909(119905) 119905) along the solutions of (34)we have

119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)

Equation (34) can be rewritten as

(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)

Letting V = 119909(119905)minusnabla119909(119909(119905) 120583(119905))minus120576(119905)nabla

119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have

⟨119909 (119905) minus nabla119909 (119909 (119905) 120583 (119905)) minus 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

minus (119905) minus 119909 (119905) (119905) + 119909 (119905) minus 119909 (119905)⟩ ge 0

(50)

which follows the fact that

⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)

Combining (48) and (51) we get that

119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)

From Lemmas 4 and 5 (119905) = minus119890minus119905 and 120576(119905) lt 0 we obtainthe estimation in (i)

(ii) Differentiating 119866(119909(119905) 119905) along the solutions of (34)we have

119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and

119906 = 119875M(119909(119905)) using (10) and (49) we have

⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)


Combining (53) and (55) we get

119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)

Similar to the analysis in (i) we obtain the estimation in(ii)

Next we prove the efficiency of proposed network (34) forsolving optimization problem (24) where the convergencefeasibility of the proposed network is a basic property

Theorem 9 For initial point 1199090isin Ω any solution 119909 [0 +infin)

rarr 119877119899 of (34) satisfies

lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)

Proof Denote 119909 [0 +infin) rarr 119877119899 as a global solution of (34)

with initial point 1199090isin Ω

From 119909(119905) isin Ω forall119905 isin [0 +infin) and Lemma 8 119864(119909(119905) 119905) isnonincreasing along the solution of (34) Using119864(119909(119905) 119905) ge 0forall119905 isin [0 +infin) we confirm that

lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)

From 119909(119905) isin Ω forall119905 isin [0 +infin) and lim119905rarr+infin

120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)

Inwhat follows wewill prove that lim119905rarr+infin

(119909(119905) 120583(119905)) =

0 Arguing by contradiction we assume that

lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)

By (119909(119905) 120583(119905)) ge 119901(119909(119905)) ge 0 forall119905 isin [0 +infin) we have 119888 gt 0which follows the fact that there is 119879

1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)

From Lemma 4 we obtain

1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)

which implies that there exists 1198792gt 119879

1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)

Since Ω is bounded and 119909(119905) isin Ω forall119905 isin [0 +infin) byLemma 5 there is 119877 gt 0 such that10038161003816100381610038161003816⟨nabla

119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877

forall119905 isin [0 +infin)

(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)

which implies that there is 1198793ge 119879

2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)

From Lemma 8 and the convexity of (sdot 120583) for any fixed120583 gt 0 we obtain that

119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)

sdot ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(68)

By (62) (67) and (68) we obtain

119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)

Integrating the above inequality from 1198793to 119905 (gt119879

3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)

which leads to a contradiction with 119866(119909 119905) ge 0 for all 119909 isin 119877119899and 119905 isin [0 +infin) Therefore

lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)

which guarantees that

lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)

The following theorem indicates that any accumulationpoint of the solutions of (34) is just an optimal solution of(34)


Theorem 10 For initial point 1199090isin Ω any solution 119909 of (34)

is convergent to the optimal solution setM that is

lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)

Proof From (63) (68) and (119909(119905) 120583(119905)) ge 0 forall119905 isin [0 +infin)we have

119889

119889119905119866 (119909 (119905) 119905)

le minus120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

+ 119898120583 (119905)

(75)

by (119905) = minus120583(119905) which can be rewritten as

119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905

isin [0 +infin) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le 0

119869 = 119905


gt 0

(77)

Owning to the continuity of ⟨nabla119909(119909(119905) 120583(119905)) 119909(119905)minus119875M(119909(119905))⟩

on [0 +infin) 119868 and 119869 are closed and open in [0 +infin) respec-tively

Case 1 In this case we assume that there exists 119879 ge 0 suchthat 119905 isin 119868 forall119905 isin [119879 +infin)

From the definition on 119868 and the pseudoconvexity of(sdot 120583) on 119877119899 we have

(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies

lim sup119905rarr+infin

119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)

ByTheorem 9 we confirm that

lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)

Case 2 In this case we assume that there exists 119879 ge 0 such

that 119905 isin 119869 forall119905 isin [119879 +infin) which means that

119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] lt 0 forall119905 isin [119879 +infin) (81)

Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)

Since lim119905rarr+infin

119864(119909(119905) 119905) exists and lim119905rarr+infin

120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose

lim inf119905rarr+infin

⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)

Then there is 1198791ge such that

⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)

Then from (76) we have

119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)

Let 119905 rarr +infin in the above inequality then we have

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)

which leads to a contraction with the boundedness frombelow 119866(119909(119905) 119905) on [0 +infin) Thus


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)

which follows the fact that there is an increasing sequence 119905119899

such that 119905119899rarr +infin and

lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)

By 119909(119905119899) sube Ω and Ω which is bounded there are 119909lowastlowast isin

Ω and a subsequence of 119909(119905119899) (denoted as 119909(119905

119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909

lowastlowast By Lemma 5there is 120585(119909lowastlowast) isin 120597119891(119909lowastlowast) such that

⟨120585 (119909lowastlowast

) 119909lowastlowast

minus 119875M (119909lowastlowast

)⟩ = 0 (91)

From the pseudoconvexity of 119891 and the above inequalitywe have

119891 (119909lowastlowast

) le 119891 (119875M (119909lowastlowast

)) (92)


Table 1 Experimental data set in MDF hot-pressing

Regression parameter 120574 120590 119862 MSE DF SEMOR 0001 5 30 00307 09543 minus08299

MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388

By Theorem 9 we have 119909lowastlowast isin F Therefore 119909lowastlowast isin Mwhich implies that

lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)

Combining the above results with (83) we conclude thatlim

119905rarr+infindist(119909(119905)M) = 0

Case 3 In this case we assume that both 119868 and 119869 areunbounded

For 119905 isin 119868 similar to the analysis in Case 1 we have

lim119905rarr+infin119905isin119868

dist (119909 (119905) M) = 0 (94)

For 119905 isin 119869 define 120591(119905) = sup119904le119905119904isin119868

119904 Then 120591(119905) isin 119868 and(120591(119905) 119905] sube 119869 By the unboundedness of 119868 and 119869 we have

lim119905isin119869119905rarr+infin

120591 (119905) = +infin (95)

From (81) and the continuity of 119866(119909(119905) 119905) on [0 +infin) wehave

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)

By 120591(119905) isin 119868 (94) and (95) we have

lim sup119905rarr+infin119905isin119869

[119866 (119909 (119905) 119905) + 119898120583 (119905)]

le lim119905rarr+infin119905isin119869

[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)

From the definition of 119866(119909 119905) and the above result wehave


dist (119909 (119905) M) = 0 (99)

Therefore from (94) and (99) we obtain

lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)

5 Numerical Experiments

In this section we test the proposed neural network (34)for solving problem (24) which is brought forward from

the MDF continuous hot-pressing process Based on theexisting data set we use the established theories and proposedneural network (34) to build the relationships between themain qualities of the hot-pressing plate and some relativetechnology parameters from optimization problem (18)Then based on optimization problem (22) we will useproposed network (34) to solve the optimal values of thetechnology parameters in hot-pressing system for optimizingthe qualities of the hot-pressing plate All these numericalexperiments validate the good performance of the proposednetwork in this paper

The numerical testing was carried out on a Lenovo PC(300GHz 200GB of RAM) with the use of Matlab 74 Andwe use ode23 to realize the neural network (34) in Matlab

51 Construction Relation Models in MDF Continuous Hot-Pressing Process In this part by considered optimizationproblem (18) and network (34) we build the relation modelwhich takes the hot-pressing temperature (TE) hot-pressingpressure (PR) hot-pressing time (TI) and moisture content(MC) of slab as the argument variables and the MOR MOEand IBS indexes of MDF as the dependent variables Thenumerical results show the good fitting of the built modelsfor the data set where the data set is given in Table 4

In order to use the data in Table 4 we first normalize theminto [0 1] And we use the mean square error (MSE) degreeof fitting (DF) and sufficient evaluation (SE) to evaluate thenumerical results where

MSE MSE = radicsum119899

119894=1(119909

119894minus 119909

lowast

119894)2

119899

DF DF = 1 minus radic(sum119899

119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)

where 119909 and 119909lowast indicate the actual value and predicted valueand 119876 and 119910 are defined as in (18) The smaller the MSE andthe SE and the closer the DF to 1 the better the regressionresult Moreover the SE function is the objective function in(15)

Based on the Gaussian radial basis function kernel thevalues of the initial parameters in problem (18) are given inTable 1 With a random initial point 119909

0isin Ω the numerical

results with respect to the obtained solution are also listed inTable 1 Figures 2ndash4 illustrate the fitting effect of the MORMOE and IBS values by using the proposed network (34) forsolving problem (18) And the parameters of the regressionfunctions in (19) are shown in Table 5


Table 2 Criterion of physical and mechanical performance indexes

Performance index Unit Thickness range4sim6 6sim9 9sim12 12sim19 19sim30 30sim45 gt45

IBSExcellent grade 065 065 060 055 055 050 050First grade Mpa 060 060 055 050 050 045 045Accepted product 055 055 050 045 045 045 045

MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45

Figure 2 Normalized sample fitting results of network (34) for theregression of MOR

Moreover the regression functions based on the linearpolynomial kernel in (20) are also calculated by network (34)where the parameters are also shown in Table 5

52 Optimization of Parameters in MDF Continuous Hot-Pressing Process In this subsection we consider two classesof parameter optimizing problems one aims at optimizingone performance index of MDF and the other is for com-prehensively optimizing the three performance indexes ofMDF Based on the obtained relation models for a particularslab and hot-pressing system we give a suggestion on thesetting of hot-pressing pressure hot-pressing temperaturehot-pressing time and moisture content of slab to let theMDF meet the given requirements We refer to the currentstandard of MDF indoor plate in China (GBT11718-1999)which is given in Table 2

521 Case 1

Case 1 When the MOR and MOE of the hot-pressing plateare in the certain regions we would like to maximize theIBS by control of the hot-pressing pressure hot-pressingtemperature hot-pressing time andmoisture content of slab

OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45

Figure 3 Normalized sample fitting results of network (34) for theregression of MOE

By the information in Table 2 we let ]1= 200 ]

2= 4

]3= 7 and ]

4= 12 and choose 119910lowast

1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576

2= 500 in optimization problem (21)

Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)

By the proposed network (34) for solving (21) we obtainthe optimal solution

119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45

Figure 4 Normalized sample fitting results of network (34) for theregression of IBS

k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215

Figure 5 Convergence of MOR regression function 1198911(119909) along the

solution of (34)

Table 3 Case 1 performance index values of obtained hot-pressingplate

MOR (Mpa) MOE (Mpa) IBS217247 26504081 06040

This means that when we let the hot-pressing temperaturebe 1594725∘C hot-pressing pressure be 31022Mpa hot-pressing time be 64866min and moisture content of slab be60061 we can maximize the IBS of the hot-pressing plateand let the MOR and MOE of it be in the certain regionswhere the three performance indexes are shown in Table 3The convergence of theMOR regression function119891

1(119909)MOE

regression function 1198912(119909) and IBS regression function 119891

3(119909)

along the solution of (34) are plotted in Figures 5ndash7

2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k

Figure 6 Convergence of MOE regression function 1198912(119909) along the

solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k

Figure 7 Convergence of IBS regression function 1198913(119909) along the

solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t

Figure 8 Convergence of 119891(119909) along the solution of (34)



Number TE (∘C) PR (MPa) TI (min) MC () MOR (Mpa) MOE (MPa) IB (Mpa)1 170 3 4 8 2566 253893 047752 170 34 5 10 291275 2887788 0623 170 27 6 12 3395 3159645 06354 180 3 6 10 299025 2967803 06255 180 34 4 12 3176 3034485 070256 180 27 5 8 331925 3215603 056257 190 3 5 12 3398 312216 0728 190 34 6 8 351575 3191608 0639 190 27 4 10 31955 3034818 06510 180 3 4 8 2513 229626 04211 170 34 4 10 2036 260156 0512 170 27 4 12 28 274305 04413 190 3 4 12 3574 306348 068714 180 34 4 8 222025 2707743 0715 190 27 5 10 3085 32127 06816 190 3 6 12 3265 298233 06617 170 34 6 10 2938 348739 07718 180 27 4 8 2587 230419 04119 190 27 6 10 2763 253905 04920 180 27 5 10 2835 267537 04221 170 27 5 8 1463 180699 0543122 180 3 5 12 3452 346212 06323 190 34 5 10 3807 345779 077

Table 5 Parameter values in the regression functions

In the regression functions (19) In the regression functions (20)120588lowast

1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282

Figure 9 Convergence ofMOR regression function1198911(119909) along the

solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840

Figure 10 Convergence ofMOE regression function1198912(119909) along the

solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2

Case 2 We would like to maximize the MOR MOE andIBS synthetically by control of the hot-pressing pressurehot-pressing temperature hot-pressing time and moisturecontent of slab

In this case we use optimization problem (23) with 1205821=

13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast

2= 2500 (Mpa) and 119891lowast

3= 060 (Mpa)

Using proposed network (34) to solve (23) we obtain theoptimal solution

119909lowast

= (1822461 30991 85167 51801)119879

(104)

The convergence of 119891(119909) along the solution of (23) is plottedin Figure 8 where the convergence of MOR regressionfunction 119891

1(119909) MOE regression function 119891

2(119909) and IBS

regression function 1198913(119909) is shown in Figures 9ndash11

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (2572014AB03) and the NationalNatural Science Foundation of China (31370565)

References

[1] M Irle and C Loxton ldquoThe manufacture and use of panelproducts in the UKrdquo Journal of the Institute of Wood Sciencevol 14 no 1 pp 21ndash26 1996

[2] D Olah R Smith and B Hansen ldquoWood material use in theUS cabinet industry 1999 to 2001rdquo Forest Products Journal vol53 no 1 pp 25ndash31 2003

[3] J J Hopfield and DW Tank ldquolsquoNeuralrsquo computation of decisonsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985

[4] M P Kennedy and L O Chua ldquoNeural networks for nonlinearprogrammingrdquo IEEE Transactions on Circuits and Systems vol35 no 5 pp 554ndash562 1988

[5] D W Tank and J J Hopfield ldquoSimple lsquoneuralrsquo optimizationnetworks an AD converter signal decision circuit and alinear programming circuitrdquo IEEE Transactions on Circuits andSystems vol 33 no 5 pp 533ndash541 1986

[6] W Bian and X P Xue ldquoSubgradient-based neural networks fornonsmooth nonconvex optimization problemsrdquo IEEE Transac-tions on Neural Networks vol 20 pp 1024ndash1038 2009

[7] W Bian and X Xue ldquoNeural network for solving constrainedconvex optimization problems with global attractivityrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 60no 3 pp 710ndash723 2013

[8] L Cheng Z-G Hou Y Z Lin M Tan W C Zhang andF-X Wu ldquoRecurrent neural network for non-smooth convexoptimization problems with application to the identificationof genetic regulatory networksrdquo IEEE Transactions on NeuralNetworks vol 22 no 5 pp 714ndash726 2011


[9] Q Liu Z Guo and J Wang ldquoA one-layer recurrent neuralnetwork for constrained pseudoconvex optimization and itsapplication for dynamic portfolio optimizationrdquo Neural Net-works vol 26 pp 99ndash109 2012

[10] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 39 no 7 pp 441ndash4521992

[11] X Hu and J Wang ldquoDesign of general projection neural net-works for solving monotone linear variational inequalities andlinear and quadratic optimization problemsrdquo IEEE Transactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no5 pp 1414ndash1421 2007

[12] Q Liu and J Cao ldquoA recurrent neural network based on pro-jection operator for extended general variational inequalitiesrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 40 no 3 pp 928ndash938 2010

[13] Y Xia G Feng and J Wang ldquoA recurrent neural network withexponential convergence for solving convex quadratic programand related linear piecewise equationsrdquoNeural Networks vol 17no 7 pp 1003ndash1015 2004

[14] Q X Zhu and J D Cao ldquoExponential stability of stochasticneural networks with both Markovian jump parameters andmixed time delaysrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 41 no 2 pp 341ndash353 2011

[15] M Forti P Nistri and M Quincampoix ldquoGeneralized neuralnetwork for nonsmooth nonlinear programming problemsrdquoIEEE Transactions on Circuits and Systems I Regular Papers vol51 no 9 pp 1741ndash1754 2004

[16] M Forti P Nistri and M Quincampoix ldquoConvergence ofneural networks for programming problems via a nonsmoothŁojasiewicz inequalityrdquo IEEE Transactions on Neural Networksvol 17 no 6 pp 1471ndash1486 2006

[17] W L Lu and J Wang ldquoConvergence analysis of a class ofnonsmooth gradient systemsrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 55 no 11 pp 3514ndash3527 2008

[18] W Bian and X Chen ldquoSmoothing neural network for con-strained non-lipschitz optimization with applicationsrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 3 pp 399ndash411 2012

[19] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983

[20] R T Rockafellar and R J Wets Variational Analysis vol 317of Grundlehren der Mathematischen Wissenschaften SpringerBerline Germany 1998

[21] B Scholkopf and A J Smola Learning with Kernels MIT PressCambridge Mass USA 2002

[22] F Liu and X Xue ldquoDesign of natural classification kernels usingprior knowledgerdquo IEEE Transactions on Fuzzy Systems vol 20no 1 pp 135ndash152 2012

[23] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[24] C Chen andO LMangasarian ldquoA class of smoothing functionsfor nonlinear and mixed complementarity problemsrdquo Compu-tational Optimization and Applications vol 5 no 2 pp 97ndash1381996

[25] J-P Penot and PHQuang ldquoGeneralized convexity of functionsand generalized monotonicity of set-valued mapsrdquo Journal ofOptimization Theory and Applications vol 92 no 2 pp 343ndash356 1997

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


projection method Then Lagrangian and penalty methodsare introduced into networks Based on the Lagrangian func-tionmethod Lagrangian networks were proposed for solvingthe optimization problems [8 10] with general constraintsBut the Lagrangian network increases the dimension of thenetworks along with the number of the constraints In recentyears recurrent neural networks based on penalty methodwere widely investigated for solving optimization problemsThe neural networks for smooth optimization problems cannot solve nonsmooth optimization problems because thegradients of the objective and constrained functions arerequired in such neural networks The generalized nonlinearprogramming circuit (G-NPC) in [15] can be considered asa natural extension of nonlinear programming circuit (NPC)for solving nonsmooth convex optimization problems withinequality constraints But the nonempty interior of feasibleregion and large enough penalty parameters are needed forthe network in [15] In order to overcome the nonemptyassumption of the interior of feasible region Bian and Xue[6] proposed a recurrent neural network for nonsmoothconvex optimization based on penalty function methodThe efficiency of the neural networks for solving convexoptimization problems relies on the convexity of functionsA neural network for nonconvex quadratic optimization ispresented in [16] Some neural networks modeled by differ-ential inclusion were also proposed for some nonsmooth andnonconvex optimization problems [6 17] To overcome thedifferential inclusion smoothing techniques are introducedinto the neural networks The main feature of smoothingmethod is to approximate the nonsmooth functions by a classof smooth functions Thus the neural network constructedby the smoothing techniques is modeled by a differentialequation which can be implemented easily in circuits andmathematical software [18]

In this paper we propose a neural network model forsolving the optimization problem brought forward from theMDF continuous hot-pressing automatic control system InSection 2 some notations and necessary preliminary resultsare listed In Section 3 based on the SVM theory withthe existing linear and nonlinear kernel functions we givean optimization problem which includes the problems forbuilding the models of MDF continuous hot-pressing systemand optimizing the MDF performance indexes as specialcases In order to build up the relation models on theproperties of the slab technical parameters in hot-pressingprocess and the performance indexes of MDF when the ker-nel function is positive definite or semipositive definite thecorresponding optimization problem is a constrained convexproblem otherwise it is a nonconvex problemThe optimiza-tion problem for optimizing the performance parametersis a nonconvex constrained optimization problem but itsobjective function is pseudoconvex due to the appropriatechoice of kernel functions In Section 4 we propose a neuralnetwork based on the penalty function method projectionmethod and smoothing techniques The proposed networkis modeled by a nonautomatic differential equation ByLyapunovmethod we prove that the solution of the proposednetwork is global existent and convergent to the feasible setof the considered optimization problems Moreover due to

the pseudoconvexity of the objective function and the con-vexity of the constraint the proposed network also convergesto the optimal solution set of the optimization problem InSection 5 based on the existing data set we use the proposednetwork into the model building and parameter optimizingproblems of hot-pressing system which validates the goodperformance of the obtained results in this paper

Notations 119877+= [0 +infin) Given column vectors 119909 = (119909

1

1199092 119909

119899)119879 and 119910 = (119910

1 119910

2 119910

119899)119879 ⟨119909 119910⟩ = 119909

119879

119910 =

sum119899

119894=1119909119894119910119894is the scalar product of 119909 and 119910 119909

119894denotes the

119894th element of 119909 119909 denotes the Euclidean 2-norm definedby 119909 = (sum

119899

119894=11199092

119894)12 For a closed convex subset Γ sube 119877

119899dist(119909 Γ) is the distance from 119909 to Γ defined by dist(119909 Γ) =min

119910isinΓ119909 minus 119910

2 Preliminaries

In this section we state some definitions and propertiesneeded in this paper We refer the readers to [19ndash21]

21 Support Vector Regression Kernels were regarded as afunction with the formulation of inner product and havebeen a powerful tool in machine learning for their superiorperformance over a wide range of learning problems suchas isolated handwritten digit recognition text categorizationand face detection [21 22]

LetX be a nonempty set and 119870 X timesX rarr 119877 be a real-valued and symmetric function With the kernel matrix K =

(119896(119909119894 119909

119895))119899

119894119895=1 119870 is said to be a positive semidefinite kernel

ifK is positive semidefinite for any 119899 isin N and 1199091 119909

2 119909

119899isin

XWe call119870 an indefinite kernel if there exist1199091 119909

2 119909

119899isin

X and V 119908 isin 119877119899 such that V119879KV lt 0 and 119908119879K119908 gt 0In what follows we list some widely used kernels

(i) Gaussian radial basis function kernel 119870(119909 119910) =

exp(minus119909 minus 119910221205902)

(ii) Polynomial kernel 119870(119909 119910) = (⟨119909 119910⟩ + 119888)119901 119888 ge 0

119901 ge 1

(iii) Sigmoid kernel 119870(119909 119910) = tanh(120581⟨119909 119910⟩ + V) 120581 gt 0V lt 0

22 Smoothing Approximation Smoothing approximation isan effective method for solving nonsmooth optimizationproblems and has been widely used in the past decades Themain feature of smoothing method is to approximate thenonsmooth functions by a class of parameterized smoothfunctions In this paper we adopt the smoothing functiondefined as follows

Definition 1 (see [23]) Let 119892 119877119899

rarr 119877 be a continuousfunction One calls 119877119899 times119877

+rarr 119877 a smoothing function of

119892 if (sdot 120583) is continuously differentiable for any fixed 120583 gt 0and lim

119911rarr119909120583darr0(119911 120583) = 119892(119909) holds for any 119909 isin 119877119899






120588 (119905) = 120588 (minus119905)

120581 fl int

infin

minusinfin

|119905| 120588 (119905) 119889119905 lt infin

(1)

Then

120601 (119905 120583) fl int

infin

minusinfin

(119904 minus 120583119905)+120588 (119905) 119889119905 (2)

from 119877 times 119877+to 119877

+is well defined



1206011(119904 120583) = 119904 + 120583 ln (1 + 119890minus119904120583)

1206012(119904 120583) =

1

2(119904 + radic1199042 + 41205832)

1206013(119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

1206014(119904 120583) =

119904 +120583

2119890minus119904120583 if 119904 gt 0

120583

2119890minus119904120583 if 119904 le 0

(3)


2is



4



|119904| = (119904)++ (minus119904)

+

max (119904 119905) = 119904 + (119905 minus 119904)+

min (119904 119905) = 119904 minus (119904 minus 119905)+


(4)


+


119899


119866(119909) = con V | nabla

119909 (119909

119896

120583119896) 997888rarr V for 119909119896 997888rarr 119909 120583

119896

darr 0

(5)



120597119892 (119909) = 119866(119909) (6)




exist120585 (119909) isin 120597120601 (119909)

st ⟨120585 (119909) 119910 minus 119909⟩ ge 0 997904rArr

120601 (119910) ge 120601 (119909)

(7)



sum

119894=1

1199092

119894

1205902

119894

) (8)



119875X (119909) = argmin119906isinX

119906 minus 1199092 (9)




(10)


then 119875Ω(119909) = (119901

1 119901

2 119901


119901119894=

119906119894119909119894lt 119906

119894

119909119894119906119894le 119909

119894le V

119894

V119894

V119894lt 119909

119894

(11)



119879

(119860119860119879

)minus1

(119860119909 minus 119887)






119901 andY sub 119877

as two sets where 119909119894 = (1199091198941 119909

119894

2 119909

119894



1

1199042

119904119899



min120572120573

1

2

119899

sum

119894119895=1

(120573119894minus 120572

119894) (120573

119895minus 120572

119895)119870 (119909

119894

119909119895

)

minus

119899

sum

119894=1

(120573119894minus 120572

119894) 119910

119894

+120576

2119862

119899

sum

119894=1

(120573119894minus 120572

119894)2

st119899

sum

119894=1

(120573119894minus 120572

119894) = 0

0 le 120573119894 120572

119894le 119862 119894 = 1 2 119899

(12)




119891 (119909) =

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894)119870 (119909

119894

119909) + 119887 (13)


119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895

⟩ + 120576 120572lowast

119895isin (0 119862)

119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895


119895isin (0 119862)

(14)

Denote 119909119894= 120573

119894minus 120572


as

min 1

2119909119879

119876119909 minus 119910119879

119909

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(15)

where 119909 = (1199091 119909

2 119909

119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1

1)119879

isin 119877119899 and 119876 = K+(1205762119862)119868

119899withK = (119870(119909

119894

119909119895

))119899

119894119895=1


lowast

2 119909

lowast

119899) of (15) we call




min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(16)

with 120574 gt 0 In (43) 120574sum119899




119875 (119909) =

1003817100381710038171003817100381710038171003817

1

119899119890119890119879

119909

1003817100381710038171003817100381710038171003817=radic119899

119899

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=1

119909119894

1003816100381610038161003816100381610038161003816100381610038161003816

(17)


min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816

st minus 119862119890 le 119909 le 119862119890

(18)



1198911(119909) fl

119899

sum

119894=1

120588lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

3(IBS)

(19)


1198911(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894⟨119911

119894 119909⟩ + 119887

3(IBS)

(20)


where 1199111 119911

2 119911


regression and 119909 = (1199091 119909

2 119909

3 119909





4



(ii) 119891119894and minus119891



119877119899 119894 = 1 2 3



(ii) 119891119894and minus119891

119894are convex on 119877119899 119894 = 1 2 3



min minus 1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le 119891

1(119909) minus 119910

lowast

1le 120576

1

minus 1205802le 119891

2(119909) minus 119910

lowast

2le 120576

2

(21)



1minus120580

1 119910

lowast

1+120576

1] [119910lowast

2minus120580

2 119910

lowast

2+




2in the


min minus

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1le 120576

1

minus 1205802le

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2le 120576

2

(22)



min minus1205821

119891lowast

1

1198911(119909) minus

1205822

119891lowast

2

1198912(119909) minus

1205823

119891lowast

3

1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

(23)

where 1205821 120582

2 120582




2 and 119891lowast



1= 120582

2= 120582


1= 12

1205822= 13 and 120582



1119891

2 and119891

3in problem (23)



minimize 119891 (119909) fl Θ (119909) + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816


119892119894(119909) le 0 119894 = 1 119898

(24)



119899

rarr 119877


119899




if we let


119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

119886 fl 0

119887 fl (]1 ]2 ]3 ]4)119879

(25)

1198921(119909) fl minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

(26)

1198923(119909) fl minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(27)




F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)


4 Main Results



119901 (119909) =

119898

sum

119894=1

max 0 119892119894(119909) (29)



(119909 120583) =

119898

sum

119894=1

120601 (119892119894(119909) 120583) (30)

where

120601 (119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

(31)





lim119904rarr119905120583darr0

nabla119904120601(119904 120583) sube 120597max0 119905







120579 (119904 120583) =

|119904| if |119904| ge 120583

1199042

2120583+120583

2if |119904| lt 120583

(32)

we define

(119909 120583) fl Θ (119909) + 120574

119899

sum

119894=1

120579 (119909119894 120583) + 120590120579 (119890

119879

119909 120583) (33)




(ii) lim119911rarr119909120583darr0

nabla119909(119909 120583) sube 120597119891(119909)




119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(34)



120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla



(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)



119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)














119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)


(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)


119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)


119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905





obtain that



0isin Ω


119909(sdot 120583) and







0isin Ω





119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120588 (119905) = 120588 (minus119905)

120581 fl int

infin

minusinfin

|119905| 120588 (119905) 119889119905 lt infin

(1)

Then

120601 (119905 120583) fl int

infin

minusinfin

(119904 minus 120583119905)+120588 (119905) 119889119905 (2)

from 119877 times 119877+to 119877

+is well defined



1206011(119904 120583) = 119904 + 120583 ln (1 + 119890minus119904120583)

1206012(119904 120583) =

1

2(119904 + radic1199042 + 41205832)

1206013(119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

1206014(119904 120583) =

119904 +120583

2119890minus119904120583 if 119904 gt 0

120583

2119890minus119904120583 if 119904 le 0

(3)


2is



4



|119904| = (119904)++ (minus119904)

+

max (119904 119905) = 119904 + (119905 minus 119904)+

min (119904 119905) = 119904 minus (119904 minus 119905)+


(4)


+


119899


119866(119909) = con V | nabla

119909 (119909

119896

120583119896) 997888rarr V for 119909119896 997888rarr 119909 120583

119896

darr 0

(5)



120597119892 (119909) = 119866(119909) (6)




exist120585 (119909) isin 120597120601 (119909)

st ⟨120585 (119909) 119910 minus 119909⟩ ge 0 997904rArr

120601 (119910) ge 120601 (119909)

(7)



sum

119894=1

1199092

119894

1205902

119894

) (8)



119875X (119909) = argmin119906isinX

119906 minus 1199092 (9)




(10)


then 119875Ω(119909) = (119901

1 119901

2 119901


119901119894=

119906119894119909119894lt 119906

119894

119909119894119906119894le 119909

119894le V

119894

V119894

V119894lt 119909

119894

(11)



119879

(119860119860119879

)minus1

(119860119909 minus 119887)






119901 andY sub 119877

as two sets where 119909119894 = (1199091198941 119909

119894

2 119909

119894



1

1199042

119904119899



min120572120573

1

2

119899

sum

119894119895=1

(120573119894minus 120572

119894) (120573

119895minus 120572

119895)119870 (119909

119894

119909119895

)

minus

119899

sum

119894=1

(120573119894minus 120572

119894) 119910

119894

+120576

2119862

119899

sum

119894=1

(120573119894minus 120572

119894)2

st119899

sum

119894=1

(120573119894minus 120572

119894) = 0

0 le 120573119894 120572

119894le 119862 119894 = 1 2 119899

(12)




119891 (119909) =

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894)119870 (119909

119894

119909) + 119887 (13)


119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895

⟩ + 120576 120572lowast

119895isin (0 119862)

119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895


119895isin (0 119862)

(14)

Denote 119909119894= 120573

119894minus 120572


as

min 1

2119909119879

119876119909 minus 119910119879

119909

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(15)

where 119909 = (1199091 119909

2 119909

119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1

1)119879

isin 119877119899 and 119876 = K+(1205762119862)119868

119899withK = (119870(119909

119894

119909119895

))119899

119894119895=1


lowast

2 119909

lowast

119899) of (15) we call




min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(16)

with 120574 gt 0 In (43) 120574sum119899




119875 (119909) =

1003817100381710038171003817100381710038171003817

1

119899119890119890119879

119909

1003817100381710038171003817100381710038171003817=radic119899

119899

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=1

119909119894

1003816100381610038161003816100381610038161003816100381610038161003816

(17)


min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816

st minus 119862119890 le 119909 le 119862119890

(18)



1198911(119909) fl

119899

sum

119894=1

120588lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

3(IBS)

(19)


1198911(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894⟨119911

119894 119909⟩ + 119887

3(IBS)

(20)


where 1199111 119911

2 119911


regression and 119909 = (1199091 119909

2 119909

3 119909





4



(ii) 119891119894and minus119891



119877119899 119894 = 1 2 3



(ii) 119891119894and minus119891

119894are convex on 119877119899 119894 = 1 2 3



min minus 1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le 119891

1(119909) minus 119910

lowast

1le 120576

1

minus 1205802le 119891

2(119909) minus 119910

lowast

2le 120576

2

(21)



1minus120580

1 119910

lowast

1+120576

1] [119910lowast

2minus120580

2 119910

lowast

2+




2in the


min minus

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1le 120576

1

minus 1205802le

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2le 120576

2

(22)



min minus1205821

119891lowast

1

1198911(119909) minus

1205822

119891lowast

2

1198912(119909) minus

1205823

119891lowast

3

1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

(23)

where 1205821 120582

2 120582




2 and 119891lowast



1= 120582

2= 120582


1= 12

1205822= 13 and 120582



1119891

2 and119891

3in problem (23)



minimize 119891 (119909) fl Θ (119909) + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816


119892119894(119909) le 0 119894 = 1 119898

(24)



119899

rarr 119877


119899




if we let


119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

119886 fl 0

119887 fl (]1 ]2 ]3 ]4)119879

(25)

1198921(119909) fl minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

(26)

1198923(119909) fl minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(27)




F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)


4 Main Results



119901 (119909) =

119898

sum

119894=1

max 0 119892119894(119909) (29)



(119909 120583) =

119898

sum

119894=1

120601 (119892119894(119909) 120583) (30)

where

120601 (119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

(31)





lim119904rarr119905120583darr0

nabla119904120601(119904 120583) sube 120597max0 119905







120579 (119904 120583) =

|119904| if |119904| ge 120583

1199042

2120583+120583

2if |119904| lt 120583

(32)

we define

(119909 120583) fl Θ (119909) + 120574

119899

sum

119894=1

120579 (119909119894 120583) + 120590120579 (119890

119879

119909 120583) (33)




(ii) lim119911rarr119909120583darr0

nabla119909(119909 120583) sube 120597119891(119909)




119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(34)



120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla



(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)



119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)














119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)


(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)


119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)


119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905





obtain that



0isin Ω


119909(sdot 120583) and







0isin Ω





119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901 andY sub 119877

as two sets where 119909119894 = (1199091198941 119909

119894

2 119909

119894



1

1199042

119904119899



min120572120573

1

2

119899

sum

119894119895=1

(120573119894minus 120572

119894) (120573

119895minus 120572

119895)119870 (119909

119894

119909119895

)

minus

119899

sum

119894=1

(120573119894minus 120572

119894) 119910

119894

+120576

2119862

119899

sum

119894=1

(120573119894minus 120572

119894)2

st119899

sum

119894=1

(120573119894minus 120572

119894) = 0

0 le 120573119894 120572

119894le 119862 119894 = 1 2 119899

(12)




119891 (119909) =

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894)119870 (119909

119894

119909) + 119887 (13)


119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895

⟩ + 120576 120572lowast

119895isin (0 119862)

119887 = 119910119895

minus

119899

sum

119894=1

(120573lowast

119894minus 120572

lowast

119894) ⟨119909

119894

119909119895


119895isin (0 119862)

(14)

Denote 119909119894= 120573

119894minus 120572


as

min 1

2119909119879

119876119909 minus 119910119879

119909

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(15)

where 119909 = (1199091 119909

2 119909

119899)119879 119910 = (1199101 1199102 119910119899)119879 119890 = (1 1

1)119879

isin 119877119899 and 119876 = K+(1205762119862)119868

119899withK = (119870(119909

119894

119909119895

))119899

119894119895=1


lowast

2 119909

lowast

119899) of (15) we call




min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816

st 119890119879

119909 = 0 minus 119862119890 le 119909 le 119862119890

(16)

with 120574 gt 0 In (43) 120574sum119899




119875 (119909) =

1003817100381710038171003817100381710038171003817

1

119899119890119890119879

119909

1003817100381710038171003817100381710038171003817=radic119899

119899

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119894=1

119909119894

1003816100381610038161003816100381610038161003816100381610038161003816

(17)


min 1

2119909119879

119876119909 minus 119910119879

119909 + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816

st minus 119862119890 le 119909 le 119862119890

(18)



1198911(119909) fl

119899

sum

119894=1

120588lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) + 119887

3(IBS)

(19)


1198911(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1(MOR)

1198912(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2(MOS)

1198913(119909) fl

119899

sum

119894=1

120581lowast

119894⟨119911

119894 119909⟩ + 119887

3(IBS)

(20)


where 1199111 119911

2 119911


regression and 119909 = (1199091 119909

2 119909

3 119909





4



(ii) 119891119894and minus119891



119877119899 119894 = 1 2 3



(ii) 119891119894and minus119891

119894are convex on 119877119899 119894 = 1 2 3



min minus 1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le 119891

1(119909) minus 119910

lowast

1le 120576

1

minus 1205802le 119891

2(119909) minus 119910

lowast

2le 120576

2

(21)



1minus120580

1 119910

lowast

1+120576

1] [119910lowast

2minus120580

2 119910

lowast

2+




2in the


min minus

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1le 120576

1

minus 1205802le

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2le 120576

2

(22)



min minus1205821

119891lowast

1

1198911(119909) minus

1205822

119891lowast

2

1198912(119909) minus

1205823

119891lowast

3

1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

(23)

where 1205821 120582

2 120582




2 and 119891lowast



1= 120582

2= 120582


1= 12

1205822= 13 and 120582



1119891

2 and119891

3in problem (23)



minimize 119891 (119909) fl Θ (119909) + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816


119892119894(119909) le 0 119894 = 1 119898

(24)



119899

rarr 119877


119899




if we let


119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

119886 fl 0

119887 fl (]1 ]2 ]3 ]4)119879

(25)

1198921(119909) fl minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

(26)

1198923(119909) fl minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(27)




F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)


4 Main Results



119901 (119909) =

119898

sum

119894=1

max 0 119892119894(119909) (29)



(119909 120583) =

119898

sum

119894=1

120601 (119892119894(119909) 120583) (30)

where

120601 (119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

(31)





lim119904rarr119905120583darr0

nabla119904120601(119904 120583) sube 120597max0 119905







120579 (119904 120583) =

|119904| if |119904| ge 120583

1199042

2120583+120583

2if |119904| lt 120583

(32)

we define

(119909 120583) fl Θ (119909) + 120574

119899

sum

119894=1

120579 (119909119894 120583) + 120590120579 (119890

119879

119909 120583) (33)




(ii) lim119911rarr119909120583darr0

nabla119909(119909 120583) sube 120597119891(119909)




119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(34)



120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla



(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)



119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)














119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)


(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)


119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)


119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905





obtain that



0isin Ω


119909(sdot 120583) and







0isin Ω





119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1199111 119911

2 119911


regression and 119909 = (1199091 119909

2 119909

3 119909





4



(ii) 119891119894and minus119891



119877119899 119894 = 1 2 3



(ii) 119891119894and minus119891

119894are convex on 119877119899 119894 = 1 2 3



min minus 1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le 119891

1(119909) minus 119910

lowast

1le 120576

1

minus 1205802le 119891

2(119909) minus 119910

lowast

2le 120576

2

(21)



1minus120580

1 119910

lowast

1+120576

1] [119910lowast

2minus120580

2 119910

lowast

2+




2in the


min minus

119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

minus 1205801le

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1le 120576

1

minus 1205802le

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2le 120576

2

(22)



min minus1205821

119891lowast

1

1198911(119909) minus

1205822

119891lowast

2

1198912(119909) minus

1205823

119891lowast

3

1198913(119909)

st 0 le 1199091le ]

1

0 le 1199092le ]

2

0 le 1199093le ]

3

0 le 1199094le ]

4

(23)

where 1205821 120582

2 120582




2 and 119891lowast



1= 120582

2= 120582


1= 12

1205822= 13 and 120582



1119891

2 and119891

3in problem (23)



minimize 119891 (119909) fl Θ (119909) + 120574

119899

sum

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + 120590

10038161003816100381610038161003816119890119879

11990910038161003816100381610038161003816


119892119894(119909) le 0 119894 = 1 119898

(24)



119899

rarr 119877


119899




if we let


119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

119886 fl 0

119887 fl (]1 ]2 ]3 ]4)119879

(25)

1198921(119909) fl minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

(26)

1198923(119909) fl minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(27)




F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)


4 Main Results



119901 (119909) =

119898

sum

119894=1

max 0 119892119894(119909) (29)



(119909 120583) =

119898

sum

119894=1

120601 (119892119894(119909) 120583) (30)

where

120601 (119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

(31)





lim119904rarr119905120583darr0

nabla119904120601(119904 120583) sube 120597max0 119905







120579 (119904 120583) =

|119904| if |119904| ge 120583

1199042

2120583+120583

2if |119904| lt 120583

(32)

we define

(119909 120583) fl Θ (119909) + 120574

119899

sum

119894=1

120579 (119909119894 120583) + 120590120579 (119890

119879

119909 120583) (33)




(ii) lim119911rarr119909120583darr0

nabla119909(119909 120583) sube 120597119891(119909)




119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(34)



120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla



(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)



119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)














119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)


(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)


119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)


119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905





obtain that



0isin Ω


119909(sdot 120583) and







0isin Ω





119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899

rarr 119877


119899




if we let


119899

sum

119894=1

120581lowast

119894exp(minus

1003817100381710038171003817119911119894 minus 1199091003817100381710038171003817

2

21205902) minus 119887

3

119886 fl 0

119887 fl (]1 ]2 ]3 ]4)119879

(25)

1198921(119909) fl minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) fl

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

(26)

1198923(119909) fl minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) fl

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(27)




F fl 119909 isin Ω 119892119894(119909) le 0 119894 = 1 2 119898 (28)


4 Main Results



119901 (119909) =

119898

sum

119894=1

max 0 119892119894(119909) (29)



(119909 120583) =

119898

sum

119894=1

120601 (119892119894(119909) 120583) (30)

where

120601 (119904 120583) =

max 0 119904 if |119904| gt 120583

(119904 + 120583)2

4120583if |119904| le 120583

(31)





lim119904rarr119905120583darr0

nabla119904120601(119904 120583) sube 120597max0 119905







120579 (119904 120583) =

|119904| if |119904| ge 120583

1199042

2120583+120583

2if |119904| lt 120583

(32)

we define

(119909 120583) fl Θ (119909) + 120574

119899

sum

119894=1

120579 (119909119894 120583) + 120590120579 (119890

119879

119909 120583) (33)




(ii) lim119911rarr119909120583darr0

nabla119909(119909 120583) sube 120597119891(119909)




119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(34)



120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla



(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)



119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)














119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)


(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)


119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)


119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905





obtain that



0isin Ω


119909(sdot 120583) and







0isin Ω





119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120583intminus1

minus1

xintminus

sumsum PΩminus

minus

++

120576intab

bba

aab

pnablax

xqnabla



(119905) isin minus119909 (119905) + 119875Ω[119909 (119905)

minus

119903

sum

119894=1

nabla119904120601 (119904 120583 (119905))

119904=119892119894(119909(119905))nabla119892

119894(119909 (119905)) minus 120576 (119905)

sdot (nablaΘ (119909 (119905)) + 120574

119899

sum

119894=1

nabla119904120579 (119904 120583)

119904=119909119894(119905)119890119894

+ 120590nabla119904120579 (119904 120583)

119904=119890119879119909(119905)

119890)]

(35)

where

nabla119904120601 (119904 120583) =

1 if 119904 gt 120583119904 + 120583

2120583if |119904| le 120583

0 if 119904 lt minus120583

nabla119904120579 (119904 120583) =

1 if |119904| ge 120583119904

120583if |119904| lt 120583

(36)



119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

120576 (119905) = minus1205762

(119905)

(119905) = minus120583 (119905)

(37)














119909(119909(119905) 120583(119905)) minus 120576(119905)nabla

119909(119909(119905)


(119905) + 119909 (119905) = 120579 (119905) 119905 isin [0 119879) (39)


119909 (119905) = 119890minus119905

1199090+ 119890

minus119905

int

119905

0

120579 (119904) 119890119904

119889119904 (40)


119909 (119905) = 119890minus119905

1199090+ (1 minus 119890

minus119905

)int

119905

0

120579 (119904)119890119904

119890119905 minus 1119889119904 (41)

By int1199050

119890119904

(119890119905





obtain that



0isin Ω


119909(sdot 120583) and







0isin Ω





119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 (119910 (119905) 120583 (119905))

10038161003816100381610038161003816

le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]


le 1198711003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817 forall119905 isin [119905 + 1]

(43)


Ωand (43) we have

119889

119889119905

1

2

1003817100381710038171003817119909 (119905) minus 119910 (119905)1003817100381710038171003817

2

= ⟨119909 (119905) minus 119910 (119905) (119905) minus (119905)⟩

le (119871 + 120576 (119905) 119871)1003817100381710038171003817119909 (119905) minus 119910 (119905)

1003817100381710038171003817

2

(44)


0isin Ω is unique


119864 (119909 119905) = (119909 120583 (119905)) + 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

119866 (119909 119905) =1

2dist2 (119909M) + (119909 120583 (119905))

+ 120576 (119905) ( (119909 120583 (119905)) minus infΩ

119891)

(45)



119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

(46)


119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(47)


119889

119889119905119864 (119909 (119905) 119905) = ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(48)


(119905) + 119909 (119905) = 119875Ω[119909 (119905) minus nabla

119909 (119909 (119905) 120583 (119905))

minus 120576 (119905) nabla119909 (119909 (119905) 120583 (119905))]

(49)


119909(119909(119905) 120583(119905)) and

119906 = 119909(119905) using (49) and (10) we have


119909 (119909 (119905) 120583 (119905))


(50)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

le minus (119905)2

forall119905 ge 0

(51)


119889

119889119905119864 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

(52)



119889

119889119905119866 (119909 (119905) 119905) = ⟨119909 (119905) minus 119875M (119909 (119905)) (119905)⟩

+ ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) (119905)⟩ + 120576 (119905)

sdot ( (119909 (119905) 120583 (119905)) minus infΩ

119891) + (nabla120583 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla120583 (119909 (119905) 120583 (119905))) (119905)

(53)


119909(119909(119905) 120583(119905)) and


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905))

+ (119905) (119905) + 119909 (119905) minus 119875M (119909 (119905))⟩ le 0

(54)


⟨nabla119909 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

119909 (119909 (119905) 120583 (119905)) (119905)⟩

+ ⟨ (119905) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

minus (119905)2

(55)



119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889

119889119905119866 (119909 (119905) 119905) le minus (119905)

2

+ 120576 (119905) (119891 (119909 (119905)) minus infΩ

119891)

+ (nabla120583 (119909 (119905) 120583 (119905)) + 120576 (119905) nabla

120583 (119909 (119905) 120583 (119905))) (119905)

minus ⟨nabla119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

(56)





lim119905rarr+infin

dist (119909 (119905) F) = 0 (57)




lim119905rarr+infin

119864 (119909 (119905) 119905) exists (58)


120576(119905) = 0 wehave

lim119905rarr+infin

120576 (119905) ( (119909 (119905) 120583 (119905)) minus infΩ

119891) = 0 (59)

which implies

lim119905rarr+infin

119864 (119909 (119905) 119905) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) (60)


(119909(119905) 120583(119905)) =


lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 119888 = 0 (61)


1ge 0 such that

(119909 (119905) 120583 (119905)) ge119888

2 forall119905 isin [119879

1 +infin) (62)


1003816100381610038161003816 (119875M (119909 (119905)) 120583 (119905)) minus 119901 (119875M (119909 (119905)))1003816100381610038161003816 le 119898120583 (119905) (63)


1such that

(119875M (119909 (119905)) 120583 (119905)) le119888

8 (64)


119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

10038161003816100381610038161003816le 119877


(65)

Then

lim119905rarr+infin

120576 (119905) ⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (66)


2such that

10038161003816100381610038161003816120576 (119905) ⟨nabla

119909 (119909 (119905) 120583 (119905)) 119875M (119909 (119905)) minus 119909 (119905)⟩

10038161003816100381610038161003816le119888

8


(67)


119889

119889119905119866 (119909 (119905) 119905) le minus ⟨nabla

119909 (119909 (119905) 120583 (119905))

+ 120576 (119905) nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩

le minus (119909 (119905) 120583 (119905)) + (119875M (119909 (119905)) 120583 (119905)) minus 120576 (119905)


(68)


119889

119889119905119866 (119909 (119905) 119905) le minus

119888

4 forall119905 isin [119879

3 +infin) (69)


3) we

have

119866 (119909 (119905) 119905) le 119866 (119909 (1198793) 119879

3) minus

119888

4(119905 minus 119879

3)


(70)

Thus

lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (71)


lim119905rarr+infin

119901 (119909 (119905)) = lim119905rarr+infin

(119909 (119905) 120583 (119905)) = 0 (72)


lim119905rarr+infin

dist (119909 (119905) F) = 0 (73)





lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












lim119905rarr+infin

dist (119909 (119905) M) = 0 (74)


119889

119889119905119866 (119909 (119905) 119905)


+ 119898120583 (119905)

(75)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)]


(76)

Denote

119868 = 119905


le 0

119869 = 119905


gt 0

(77)





(119909 (119905) 120583 (119905)) le (119875M (119909 (119905)) 120583 (119905))


(78)

which implies


119891 (119909 (119905)) le min119909isinF

119891 (119909) (79)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (80)



119889


Then

lim119905rarr+infin

[119866 (119909 (119905) 119905) + 119898120583 (119905)] exists (82)



120583(119905) = 0 weobtain that

lim119905rarr+infin

dist (119909 (119905) M) exists (83)

Suppose


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 119889

gt 0

(84)


⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ ge

119889

2


(85)


119889

119889119905[119866 (119909 (119905) 119905) + 119898120583 (119905)] le minus

119889

2120576 (119905)


(86)


1) we

have

119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (1198791) 119879

1) + 119898120583 (119879

1)

+ int

119905

1198791

minus119889

2120576 (119904) 119889119904

(87)


lim119905rarr+infin

119866 (119909 (119905) 119905) = minusinfin (88)



⟨nabla119909 (119909 (119905) 120583 (119905)) 119909 (119905) minus 119875M (119909 (119905))⟩ = 0 (89)



lim119899rarr+infin

⟨nabla119909 (119909 (119905

119899) 120583 (119905

119899)) 119909 (119905

119899) minus 119875M (119909 (119905

119899))⟩

= 0

(90)



119899119896)) such that

lim119896rarr+infin

119905119899119896= +infin and lim

119896rarr+infin119909(119905

119899119896) = 119909





)⟩ = 0 (91)




)) (92)




MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












MOE 0001 6 40 00642 09063 minus17060

IBS 0001 5 40 00451 09256 minus06388


lim119896rarr+infin

dist (119909 (119905119899119896) M) = 0 (93)


119905rarr+infindist(119909(119905)M) = 0




dist (119909 (119905) M) = 0 (94)




120591 (119905) = +infin (95)


119866 (119909 (119905) 119905) + 119898120583 (119905) le 119866 (119909 (120591 (119905)) 120591 (119905))

+ 119898120583 (120591 (119905)) forall119905 isin 119869

(96)



[119866 (119909 (119905) 119905) + 119898120583 (119905)]


[119866 (119909 (120591 (119905)) 120591 (119905)) + 119898120583 (120591 (119905))] = 0

(97)

which gives


119866 (119909 (119905) 119905) = 0 (98)



dist (119909 (119905) M) = 0 (99)


lim119905rarr+infin

dist (119909 (119905) M) = 0 (100)








119894=1(119909

119894minus 119909

lowast

119894)2

119899


119894=1(119909

119894minus 119909

lowast

119894)2

sum119899

119894=11199092

119894

)

SE SE = 12119909119879

119876119909 minus 119910119879

119909

(101)









MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













MOR Mpa 23 23 22 20 18 17 15MOE Mpa 2700 2700 2500 2200 2100 1900 1700

2520151050

OriginalPredicted

0

05

1

15

2

25

3

35

4

5

45




521 Case 1


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45



2= 4

]3= 7 and ]


1= 22 119910lowast

2= 2500 120580

1= 8

1205761= 8 120580

2= 500 and 120576


Then Ω = 119909 isin 1198774

0 le 1199091le ]

1 0 le 119909

2le ]

2 0 le 119909

3le

]3 0 le 119909

4le ]

4 and we can let

1198921(119909) = minus

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ minus 119887

1+ 119910

lowast

1minus 120580

1

1198922(119909) =

119899

sum

119894=1

120588lowast

119894⟨119911

119894 119909⟩ + 119887

1minus 119910

lowast

1minus 120576

1

1198923(119909) = minus

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ minus 119887

2+ 119910

lowast

2minus 120580

2

1198924(119909) =

119899

sum

119894=1

120579lowast

119894⟨119911

119894 119909⟩ + 119887

2minus 119910

lowast

2minus 120576

2

(102)


119909lowast

= (1594725 31022 64866 60061)119879

(103)


OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










OriginalPredicted

25201510500

05

1

15

2

25

3

35

4

5

45


k

454035302520151050

265

26

255

25

245

24

235

23

225

22

215


solution of (34)




1(119909)MOE


3(119909)


2750

2700

2650

2600

2550

25005 10 15 20 25 30 35 40 450

k


solution of (34)

062

06

058

056

054

052

05

0485 10 15 20 25 30 35 40 450

k


solution of (34)

10 20 30 40 50 600

1135

114

1145

115

1155

116

1165

117

1175

t







1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1minus01385 120579

lowast

1minus00335 120581

lowast

1minus0073 120588

lowast

125815 120579

lowast

1137581 120581

lowast

1minus05275

120588lowast

200139 120579

lowast

2minus00794 120581

lowast

2minus00060 120588

lowast

211358 120579

lowast

2minus07298 120581

lowast

2minus65132

120588lowast

301869 120579

lowast

301332 120581

lowast

300259 120588

lowast

324638 120579

lowast

310424 120581

lowast

397535

120588lowast

400119 120579

lowast

4minus00365 120581

lowast

4minus00000 120588

lowast

4minus10246 120579

lowast

4minus13607 120581

lowast

4minus59500

120588lowast

500948 120579

lowast

500554 120581

lowast

500939 120588

lowast

501880 120579

lowast

5minus06708 120581

lowast

563092

120588lowast

602217 120579

lowast

605229 120581

lowast

600218 120588

lowast

648889 120579

lowast

661594 120581

lowast

6155774

120588lowast

701409 120579

lowast

700001 120581

lowast

700823 120588

lowast

7minus09675 120579

lowast

7minus12267 120581

lowast

7103006

120588lowast

802361 120579

lowast

801338 120581

lowast

800054 120588

lowast

816596 120579

lowast

8minus01886 120581

lowast

8minus239383

120588lowast

9500000 120579

lowast

9500000 120581

lowast

9458063 120588

lowast

912497 120579

lowast

930084 120581

lowast

9308681

120588lowast

10minus01413 120579

lowast

10minus02405 120581

lowast

10minus01362 120588

lowast

1000644 120579

lowast

10minus22099 120581

lowast

10minus259216

120588lowast

11minus03874 120579

lowast

11minus01625 120581

lowast

11minus01190 120588

lowast

1136153 120579

lowast

11minus16600 120581

lowast

11minus270340

120588lowast

12minus00627 120579

lowast

12minus01128 120581

lowast

12minus01673 120588

lowast

12minus08880 120579

lowast

1207098 120581

lowast

12minus178497

120588lowast

1302501 120579

lowast

1300338 120581

lowast

1300488 120588

lowast

1314724 120579

lowast

13minus01911 120581

lowast

13136398

120588lowast

14minus03013 120579

lowast

14minus00673 120581

lowast

1401250 120588

lowast

14minus25732 120579

lowast

1400263 120581

lowast

14285659

120588lowast

15434098 120579

lowast

15500000 120581

lowast

15500000 120588

lowast

15minus09224 120579

lowast

1528766 120581

lowast

15267062

120588lowast

1601186 120579

lowast

1600058 120581

lowast

1600221 120588

lowast

16minus31735 120579

lowast

16minus44727 120581

lowast

16minus174470

120588lowast

17minus00048 120579

lowast

1703582 120581

lowast

1701597 120588

lowast

1700304 120579

lowast

1732063 120581

lowast

17211381

120588lowast

18minus01428 120579

lowast

18minus03514 120581

lowast

18minus01356 120588

lowast

1810775 120579

lowast

18minus08331 120581

lowast

18minus127596

120588lowast

19minus500000 120579

lowast

19minus500000 120581

lowast

19minus458399 120588

lowast

19minus16929 120579

lowast

19minus17854 120581

lowast

19minus119152

120588lowast

20minus434157 120579

lowast

20minus500000 120581

lowast

20minus500000 120588

lowast

20minus24732 120579

lowast

20minus23246 120581

lowast

20minus435023

120588lowast

21minus06476 120579

lowast

21minus07696 120581

lowast

21minus00472 120588

lowast

21minus058057 120579

lowast

21minus63103 120581

lowast

21198815

120588lowast

2201968 120579

lowast

22minus03108 120581

lowast

22minus00101 120588

lowast

2214923 120579

lowast

2233107 120581

lowast

22minus36604

120588lowast

2303606 120579

lowast

2303021 120581

lowast

2301471 120588

lowast

2330559 120579

lowast

2322478 120581

lowast

23142783

1198871

06361 1198872

06554 1198873

06052 1198871

02784 1198872

02693 1198873

04230


5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30 35 40 45 500

k

294

292

29

288

286

284

282


solution of (34)

5 10 15 20 25 30 35 40 45 500

2930

2920

2910

2900

2890

2880

2870

2860

2850

2840


solution of (34)

0608

0606

0604

0602

06

0598

0596

0594

0592

059

05885 10 15 20 25 30 35 40 45 500

k


solution of (34)

522 Case 2



13 1205822= 13 120582

3= 13 ]

1= 200 ]

2= 4 ]

3= 7 ]

4= 12

119891lowast

1= 22 (Mpa) 119891lowast


3= 060 (Mpa)


119909lowast

= (1822461 30991 85167 51801)119879

(104)



2(119909) and IBS


Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Model Building and Optimization Analysis of MDF Continuous …downloads.hindawi.com/journals/mpe/2016/1327235.pdf · 2019. 7. 30. · for solving nonsmooth convex