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Research ArticleGlobal Minimization of Nonsmooth Constrained GlobalOptimization with Filled Function
Wei-xiang Wang1 You-lin Shang2 and Ying Zhang3
1 Department of Mathematics Shanghai Second Polytechnic University Shanghai 201209 China2Department of Mathematics Henan University of Science and Technology Luoyang 471003 China3Department of Mathematics Zhejiang Normal University Jinhua 321004 China
Correspondence should be addressed to Wei-xiang Wang zhesx126com
Received 17 May 2014 Accepted 29 August 2014 Published 25 September 2014
Academic Editor Sergio Preidikman
Copyright copy 2014 Wei-xiang Wang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A novel filled function is constructed to locate a global optimizer or an approximate global optimizer of smooth or nonsmoothconstrained global minimization problems The constructed filled function contains only one parameter which can be easilyadjusted during the minimization The theoretical properties of the filled function are discussed and a corresponding solutionalgorithm is proposed The solution algorithm comprises two phases local minimization and filling The first phase minimizes theoriginal problem and obtains one of its local optimizers while the second phase minimizes the constructed filled function andidentifies a better initial point for the first phase Some preliminary numerical results are also reported
1 Introduction
Science and economics rely on the increasing demand forlocating the global optimization optimizer and thereforeglobal optimization has become one of the most attractiveresearch areas in optimization However the existence ofmultiple local minimizers that differ from the global solutionconfronts us with two difficult issues that is how to escapefrom a localminimizer to a smaller one and how to verify thatthe current minimizer is a global one These two issues makemost of global optimization problems unsolvable directly byclassical local optimization algorithms Up to now variouskinds of new theories and algorithms on global optimizationhave been presented [1ndash3] In general global optimizationmethods can be divided into two categories stochasticmethods and deterministic methodsThe stochastic methodsare usually probability based approaches such as geneticalgorithm and simulated annealing methodThese stochasticmethods have their advantages but their shortages are alsoobvious such as being easily trapped in a local optimizerDeterministic methods such as filled function method [4ndash7]
tunneling method [8] and stretching function method [9]can however often skip from the current local minimizer toa better one
The filled function approach initially proposed forsmooth optimization by Ge and Qin [4] and improved in[5ndash7] is one of the effective global optimization approachesIt furnishes us with an efficient way to use any local opti-mization procedure to solve global optimization problemThe essence of filled function method is to construct afilled function and then to minimize it to obtain a betterinitial point for the minimization of the original prob-lem The existing filled function methods are usually onlysuitable for unconstrained optimization problem More-over it requires the objective function to be continuouslydifferentiable and the number of local minimizers to befinite But in practice optimization problems may be non-smooth and often have many complicated constraints andthe number of local minimizers may also be infinite Todeal with such situation in this paper we extend filledfunction methods to the case of nonsmooth constrainedglobal optimization and propose a new filled function
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 563860 5 pageshttpdxdoiorg1011552014563860
2 Mathematical Problems in Engineering
method The proposed filled function method combinesfilled functionmethod for unconstrained global optimizationwith the exterior penalty function method for constrainedoptimization
The paper is organized as follows In Section 2 a newfilled function is proposed and its properties are discussedIn Section 3 a corresponding filled function algorithm isdesigned and numerical experiments are performed Finallyin Section 4 some conclusive remarks are given
In the rest of this paper the generalized gradient of anonsmooth function 119891(119909) at the point 119909 isin 119883 is denotedby 120597119891(119909) and the generalized directional derivative of 119891(119909)
in the direction 119889 at the point 119909 is denoted by 1198910(119909 119889) Theinterior the boundary and the closure of the set 119878 are denotedby int 119878 120597119878 and 119888119897119878 respectively
2 A New One-Parameter FilledFunction and Its Properties
21 Problem Formulation Consider the following nons-mooth constrained global optimization problem (119875)
min119909isin119878
119891 (119909) (1)
where 119878 = 119909 isin 119883 | 119892119894(119909) le 0 119894 isin 119868 119883 sub 119877119899 is a
box set 119891(119909) and 119892119894(119909) 119894 isin 119868 are Lipschitz continuous with
constants 119871119891and 119871
119892119894 119894 isin 119868 respectively and 119868 = 1 119898
is an index set For simplicity the set of local minimizersfor problem (119875) is denoted by 119871(119875) and the set of the globalminimizers is denoted by 119866(119875)
To proceed we assume that the number of minimizers ofproblem (119875) is infinite but the number of different functionvalues at the minimizers is finite
Definition 1 A function 119875(119909 119909lowast) is called a filled function of119891(119909) at a local minimizer 119909lowast if it satisfies the following
(1) 119909lowast is a strictly local maximizer of 119875(119909 119909lowast) on 119883
(2) for any 119909 isin 1198781119909lowast or 119909 isin 119883 119878 one has 0 notin 120597119875(119909 119909lowast)
where 1198781= 119909 isin 119878 | 119891(119909) ge 119891(119909lowast)
(3) if 1198782= 119909 isin 119878 | 119891(119909) lt 119891(119909lowast) is not empty then there
exists a point 1199092isin 1198782such that 119909
2is a local minimizer
of 119875(119909 119909lowast)
Definition 1 guarantees that when any local search proce-dure for unconstrained optimization is used to minimize theconstructed filled function the sequences of iterative pointwill not stop at any point at which the objective functionvalue is larger than 119891(119909lowast) If 119909lowast is not a global minimizerthen a point 119909 with 119891(119909) lt 119891(119909lowast) could be identified in theprocess of the minimization of 119875(119909 119909lowast) Then we can get abetter local minimizer of 119891(119909) by using 119909 as an initial pointBy repeating these two steps we could finally obtain a globalminimizer or an approximate globalminimizer of the originalproblem
22 A New Filled Function and Its Properties We propose inthis section a one-parameter filled function as follows
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
+ 119902[min (0max (119891 (119909) minus 119891 (119909lowast) 119892119894(119909) 119894 isin 119868))]
3
(2)
where 119902 gt 0 is a parameter and 119909lowast is the current localminimizer of 119891(119909)
Theorems 2ndash4 show that when parameter 119902 gt 0 is suitablylarge the function 119875(119909 119909lowast 119902) is a filled function
Theorem 2 Assume that 119909lowast isin 119871(119875) then 119909lowast is a strictly localmaximizer of 119875(119909 119909lowast 119902)
Proof Since 119909lowast isin 119871(119875) there is a neighborhood 119874(119909lowast 120575) of119909lowast with 120575 gt 0 such that 119891(119909) ge 119891(119909lowast) and 119892
119894(119909) le 0 119894 isin 119868
for all 119909 isin 119874(119909lowast 120575) cap 119878 We consider the following two cases
Case 1 For all 119909 isin 119874(119909lowast 120575) cap 119878 and 119909 = 119909lowast we havemin(0max(119891(119909) minus 119891(119909lowast) 119892
119894(119909) 119894 isin 119868)) = 0 and so
119875 (119909 119909lowast 119902) = minus1
119902(119891 (119909) minus 119891 (119909lowast))
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
lt minus arg 1 = 119875 (119909lowast 119909lowast 119902)
(3)
Case 2 For all 119909 isin 119874(119909lowast 120575) cap (119883 119878) there exists at least oneindex 119894
0isin 119868 such that 119892
1198940(119909) ge 0 it follows that
min (0max (119891 (119909) minus 119891 (119909lowast) 119892119894(119909) 119894 isin 119868)) = 0 (4)
Thus
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
lt minus arg 1 = 119875 (119909lowast 119909lowast 119902)
(5)
The above discussion indicates that 119909lowast is a strictly localmaximizer of 119875(119909 119909lowast 119902)
Theorem 3 Assuming that 119909lowast isin 119871(119875) then for any 119909 isin 1198781
119909 = 119909lowast or 119909 isin 119883119878 it holds that 0 notin 120597119875(119909 119909lowast1 119902) when 119902 gt 0
is big enough
Mathematical Problems in Engineering 3
Proof For any 119909 isin 1198781 119909 = 119909lowast or 119909 isin 119883 119878 we have
min(0max(119891(119909) minus 119891(119909lowast) 119892119894(119909) 119894 isin 119868)) = 0 and so
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
(6)
Thus
120597119875 (119909 119909lowast 119902)
sub minus1
119902[119891 (119909) minus 119891 (119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
times (120597119891 (119909) +119898
sum119894=1
120582119894120597119892119894(119909))
minus2 (119909 minus 119909lowast)
1 + (1 + 119909 minus 119909lowast2)2
(7)
where 0 le 120582119894le 1 119894 isin 119868Therefore when 119902 gt 0 is big enough
it holds that
⟨120597119875 (119909 119909lowast 119902) 119909 minus 119909lowast
119909 minus 119909lowast⟩
le1
119902[119871119863 +
119898
sum119894=1
max119909isin119883
1003817100381710038171003817119892119894 (119909)1003817100381710038171003817][119871
119891+119898
sum119894=1
119871119892119894]
minus21003817100381710038171003817119909 minus 119909lowast
1003817100381710038171003817
1 + (1 + 119909 minus 119909lowast2)2
lt 0
(8)
which implies that 0 notin 120597119875(119909 119909lowast 119902)
Theorem 4 Assume that 119909lowast isin 119871(119875) but 119909lowast notin 119866(119875) and119888119897 int 119878 = 119888119897119878 If 119902 gt 0 is suitably large then there exists a point1199090isin 119878 such that 119909
0is a local minimizer of 119875(119909 119909lowast 119902)
Proof By the conditions there exists an 1199092isin int 119878 such that
119891(1199092) lt 119891(119909lowast) 119892
119894(1199092) lt 0 119894 isin 119868
Then for any 119909 isin 120597119878
119875 (119909 119909lowast 119902) = minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
119875 (1199092 119909lowast 119902) = minus
1
119902[119891(1199092) minus 119891(119909lowast)]
2
minus arg (1 +10038171003817100381710038171199092 minus 119909lowast
10038171003817100381710038172)
+ 119902[max(119891(1199092) minus 119891(119909lowast) 119892
119894(1199092) 119894 isin 119868)]
3
(9)
Since as 119902 rarr +infin
119875 (119909 119909lowast 119902) ge minus120587
2 119875 (1199092 119909lowast 119902) 997888rarr minusinfin (10)
then when 119902 gt 0 is suitably big we have
119875 (1199092 119909lowast 119902) lt 119875 (119909 119909lowast 119902) forall119909 isin 120597119878 (11)
Assume that the function 119875(119909 119909lowast 119902) reaches its global mini-mizer over 119878 at 119909
0 Since 119878120597119878 is an open set then 119909
0isin 119878120597119878
and it holds that
min119909isin119878
119875 (119909 119909lowast 119902) = min119909isin119878120597119878
119875 (119909 119909lowast 119902)
= 119875 (1199090 119909lowast 119902) le 119875 (119909
2 119909lowast 119902)
(12)
In the following we will prove that
119891 (1199090) lt 119891 (119909lowast) 119892
119894(1199090) lt 0 119894 isin 119868 (13)
which leads to the resultThe proof is by contradiction Suppose that
119891 (1199090) ge 119891 (119909lowast) 119892
119894(1199090) lt 0 119894 isin 119868 (14)
Then as 119902 rarr +infin
119875 (1199090 119909lowast 119902) 997888rarr minus arg (1 +
10038171003817100381710038171199090 minus 119909lowast10038171003817100381710038172) gt minusinfin (15)
which implies that when 119902 gt 0 is suitably big we have
119875 (1199090 119909lowast 119902) gt 119875 (119909
2 119909lowast 119902) (16)
This is a contradiction
3 Algorithm and Numerical Examples
Based on the properties of the proposed filled function wenowgive a corresponding filled function algorithmas follows
31 Filled Function Algorithm FFAM
Initialization Step
(1) Set a disturbance constant 120572 = 01(2) Select an upper bound of 119902denoted by 119902
119906and set 119902
119906=
108(3) Select directions 119890
119896 119896 = 1 2 119897 with integer 119897 =
2119899 where 119899 is the number of variables(4) Set 119896 = 1
Main Step
(1) Start from an initial point 119909 minimize the problem(119875) by implementing a nonsmooth local search pro-cedure and obtain the first localminimizer119909lowast
1of119891(119909)
(2) Let 119902 = 1
4 Mathematical Problems in Engineering
(3) Construct a filled function at 119909lowast1
119875 (119909 119909lowast1 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast
1) +119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
1
10038171003817100381710038172)
+ 119902[min (0max (119891 (119909) minus 119891 (119909lowast1) 119892119894(119909) 119894 isin 119868))]
3
(17)
(4) If 119896 gt 119897 then go to (7) Else set 119909 = 119909lowast1
+ 120572119890119896as an
initial point minimize 119875(119909 119909lowast 119902) by implementing anonsmooth local search algorithm and obtain a localminimizer 119909
119896
(5) If 119909119896isin 119878 then set 119896 = 119896 + 1 and go to (4) Else go to
(6)(6) If 119909
119896meets 119891(119909
119896) lt 119891(119909lowast
1) then set 119909 = 119909
119896and
119896 = 1 Start from 119909 as a new initial point minimizethe problem (119875) by using a local search algorithmand obtain another local minimizer 119909lowast
2of 119891(119909) with
119891(119909lowast2) lt 119891(119909lowast
1) Set 119909lowast
1= 119909lowast2and go to (2) Else go to
(7)(7) Increase 119902 by setting 119902 = 10119902(8) If 119902 le 119902
119906 then set 119896 = 1 and go to (3) Else the
algorithm stops and 119909lowast1is taken as a global minimizer
of the problem (119875)
At the end of this section we make a few remarks below(1) Algorithm FFAM is comprised of two stages local
minimization and filling In stage 1 a local minimizer 119909lowast1
of 119891(119909) is identified In stage 2 filled function 119875(119909 119909lowast1 119902) is
constructed and then minimized Stage 2 terminates whenone point 119909
119896isin 1198782is located Then algorithm FFAM
reenters into stage 1 with 119909119896as an initial point to find a new
minimizer 119909lowast2of 119891(119909) (if such one minimizer exists) and so
onThe above process is repeated until some certain specifiedstopping criteria are met and then the last local minimizer isregarded as a global minimizer
(2)Themotivation andmechanism behind the algorithmFFAM are given below
In Step (3) of the Initialization Step we can choosedirections 119890
119896 119896 = 1 2 119897 as positive and negative unit
coordinate vectors For example when 119899 = 2 the directionscan be chosen as (1 0) (0 1) (minus1 0) and (0 minus1)
In Step (1) and Step (6) of the Main Step we can obtain alocal optimizer of the problem (119875) by using any nonsmoothconstrained local optimization procedure such as Bundlemethods and Powellrsquos method In Step (4) of the MainStep we can minimize the proposed filled function by usingHybrid Hooke and Jeeves-Direct Method for NonsmoothOptimization [10] Mesh Adaptive Direct Search Algorithmsfor Constrained Optimization [11] and so forth
(3) The proposed filled function algorithm can also beapplied to smooth constrained optimization Any smoothlocal minimization procedure in the minimization phase
Table 1 Computational results with initial point (minus1 minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1 minus1) (minus19802 minus00132 minus00006) minus194102 (11931 06332 minus11932) (19889 minus00001 minus00111) minus59477
can be used such as conjugate gradient method and quasi-Newton method
32 Numerical Examples Weperform the numerical tests forthree examples All the numerical tests are programmed inFortran 95 In nonsmooth case we search for the local mini-mizers by using Hybrid Hooke and Jeeves-Direct Method forNonsmooth Optimization [10] and theMesh Adaptive DirectSearch Algorithms for Constrained Optimization [11] Insmooth case we use penalty function method and conjugategradient method to get the local minimizers
The following are the three examples and their numericalresults And the symbols used in the tables are explainedbelow
119896 the iteration number in finding the 119896th localminimizer119909119896 the 119896th new initial point in finding the 119896th local
minimizer119909lowast119896 the 119896th local minimizer
119891(119909lowast119896) the function value of the 119896th local minimizer
Problem 1 Consider
min 119891 (119909) = minus11990921+ 11990922+ 11990923minus 1199091
st 11990921+ 11990922+ 11990923minus 4 le 0
min 1199092minus 1199093 1199093 le 0
(18)
Algorithm FFAM succeeds in finding an approximate globalminimizer 119909lowast = (19889 minus00001 minus00111)119879 with 119891(119909lowast) =minus59477 The computational results are given in Table 1
Problem 2 Consider
min 119891 (119909) = minus20 exp(minus02radic10038161003816100381610038161199091
1003816100381610038161003816 +10038161003816100381610038161199092
10038161003816100381610038162
)
minus exp(cos (2120587119909
1) + cos (2120587119909
2)
2) + 20
st 11990921+ 11990922le 300
21199091+ 1199092le 4
minus 30 le 119909119894le 30 119894 = 1 2
(19)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (0 0)119879 with 119891(119909lowast) = minus27183 The computationalresults are listed in Table 2
Mathematical Problems in Engineering 5
Table 2 Computational results with initial point (minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1) (minus150000 00000) 571642 (minus10585 05165) (00001 minus02094) minus036903 (00007 minus00435) (00000 00000) minus27183
Table 3 Computational results with initial point (0 0)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (0 0) (06116 34423) minus405412 (21653 22546) (23295 31780) minus55081
Problem 3 Consider
min 119891 (119909) = minus1199091minus 1199092
st 1199092le 211990941minus 811990931+ 811990921+ 2
1199092le 411990941minus 321199093
1+ 881199092
1minus 96119909
1+ 36
0 le 1199091le 3
0 le 1199092le 4
(20)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (23295 31780)119879 with 119891(119909lowast) = minus55081 This problemis taken from [12] We give this numerical example here toillustrate that algorithm FFAM is also suitable for smoothconstrained global optimization The computational resultsare given in Table 3
4 Conclusion
In this paper we present a new filled function for bothnonsmooth and smooth constrained global optimizationand investigate its properties The filled function containsonly one parameter which can be readily adjusted in theprocess of minimization We also design a correspondingfilled function algorithm Moreover in order to demonstratethe performance of the proposed filled function method wemake three numerical tests The preliminary computationalresults show that the proposed filled function approach ispromising
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thisworkwas supported by theNNSF of China (nos 11471102and 11001248) the SEDF under Grant no 12YZ178 theKey Discipline ldquoApplied Mathematicsrdquo of SSPU under noA30XK1322100 and NNSF of Zhejiang (no LY13A010006)
References
[1] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization Volume 22 Heuristic Approaches Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization vol 62 of Nonconvex Optimization and Its Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 2002
[3] R Horst and P M Pardalos Eds Handbook of Global Opti-mization vol 2 ofNonconvex Optimization and Its ApplicationsKluwer Academic Publishers Dordrecht The Netherlands1995
[4] R P Ge and Y F Qin ldquoA class of filled functions for findingglobal minimizers of a function of several variablesrdquo Journal ofOptimization Theory and Applications vol 54 no 2 pp 241ndash252 1987
[5] W Wang Y Yang and L Zhang ldquoUnification of filled functionand tunnelling function in global optimizationrdquo Acta Mathe-maticaeApplicatae Sinica English Series vol 23 no 1 pp 59ndash662007
[6] Z Xu H Huang P M Pardalos and C Xu ldquoFilled functionsfor unconstrained global optimizationrdquo Journal of Global Opti-mization vol 20 no 1 pp 49ndash65 2001
[7] Y-J Yang and Y-L Shang ldquoA new filled function method forunconstrained global optimizationrdquo Applied Mathematics andComputation vol 173 no 1 pp 501ndash512 2006
[8] A V Levy and A Montalvo ldquoThe tunneling algorithm for theglobal minimization of functionsrdquo SIAM Journal on Scientificand Statistical Computing vol 6 no 1 pp 15ndash29 1985
[9] Y Wang and J Zhang ldquoA new constructing auxiliary functionmethod for global optimizationrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1396ndash1410 2008
[10] C J Price B L Robertson and M Reale ldquoA hybrid Hookeand Jeevesmdashdirect method for non-smooth optimizationrdquoAdvanced Modeling and Optimization vol 11 no 1 pp 43ndash612009
[11] C Audet and J Dennis ldquoMesh adaptive direct search algorithmsfor constrained optimizationrdquo SIAM Journal on Optimizationvol 17 no 1 pp 188ndash217 2006
[12] C A Floudas and P M Pardalos A Collection of Test Prob-lems for Constrained Global Optimization Algorithms SpringerBerlin Germany 1990
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2 Mathematical Problems in Engineering
method The proposed filled function method combinesfilled functionmethod for unconstrained global optimizationwith the exterior penalty function method for constrainedoptimization
The paper is organized as follows In Section 2 a newfilled function is proposed and its properties are discussedIn Section 3 a corresponding filled function algorithm isdesigned and numerical experiments are performed Finallyin Section 4 some conclusive remarks are given
In the rest of this paper the generalized gradient of anonsmooth function 119891(119909) at the point 119909 isin 119883 is denotedby 120597119891(119909) and the generalized directional derivative of 119891(119909)
in the direction 119889 at the point 119909 is denoted by 1198910(119909 119889) Theinterior the boundary and the closure of the set 119878 are denotedby int 119878 120597119878 and 119888119897119878 respectively
2 A New One-Parameter FilledFunction and Its Properties
21 Problem Formulation Consider the following nons-mooth constrained global optimization problem (119875)
min119909isin119878
119891 (119909) (1)
where 119878 = 119909 isin 119883 | 119892119894(119909) le 0 119894 isin 119868 119883 sub 119877119899 is a
box set 119891(119909) and 119892119894(119909) 119894 isin 119868 are Lipschitz continuous with
constants 119871119891and 119871
119892119894 119894 isin 119868 respectively and 119868 = 1 119898
is an index set For simplicity the set of local minimizersfor problem (119875) is denoted by 119871(119875) and the set of the globalminimizers is denoted by 119866(119875)
To proceed we assume that the number of minimizers ofproblem (119875) is infinite but the number of different functionvalues at the minimizers is finite
Definition 1 A function 119875(119909 119909lowast) is called a filled function of119891(119909) at a local minimizer 119909lowast if it satisfies the following
(1) 119909lowast is a strictly local maximizer of 119875(119909 119909lowast) on 119883
(2) for any 119909 isin 1198781119909lowast or 119909 isin 119883 119878 one has 0 notin 120597119875(119909 119909lowast)
where 1198781= 119909 isin 119878 | 119891(119909) ge 119891(119909lowast)
(3) if 1198782= 119909 isin 119878 | 119891(119909) lt 119891(119909lowast) is not empty then there
exists a point 1199092isin 1198782such that 119909
2is a local minimizer
of 119875(119909 119909lowast)
Definition 1 guarantees that when any local search proce-dure for unconstrained optimization is used to minimize theconstructed filled function the sequences of iterative pointwill not stop at any point at which the objective functionvalue is larger than 119891(119909lowast) If 119909lowast is not a global minimizerthen a point 119909 with 119891(119909) lt 119891(119909lowast) could be identified in theprocess of the minimization of 119875(119909 119909lowast) Then we can get abetter local minimizer of 119891(119909) by using 119909 as an initial pointBy repeating these two steps we could finally obtain a globalminimizer or an approximate globalminimizer of the originalproblem
22 A New Filled Function and Its Properties We propose inthis section a one-parameter filled function as follows
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
+ 119902[min (0max (119891 (119909) minus 119891 (119909lowast) 119892119894(119909) 119894 isin 119868))]
3
(2)
where 119902 gt 0 is a parameter and 119909lowast is the current localminimizer of 119891(119909)
Theorems 2ndash4 show that when parameter 119902 gt 0 is suitablylarge the function 119875(119909 119909lowast 119902) is a filled function
Theorem 2 Assume that 119909lowast isin 119871(119875) then 119909lowast is a strictly localmaximizer of 119875(119909 119909lowast 119902)
Proof Since 119909lowast isin 119871(119875) there is a neighborhood 119874(119909lowast 120575) of119909lowast with 120575 gt 0 such that 119891(119909) ge 119891(119909lowast) and 119892
119894(119909) le 0 119894 isin 119868
for all 119909 isin 119874(119909lowast 120575) cap 119878 We consider the following two cases
Case 1 For all 119909 isin 119874(119909lowast 120575) cap 119878 and 119909 = 119909lowast we havemin(0max(119891(119909) minus 119891(119909lowast) 119892
119894(119909) 119894 isin 119868)) = 0 and so
119875 (119909 119909lowast 119902) = minus1
119902(119891 (119909) minus 119891 (119909lowast))
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
lt minus arg 1 = 119875 (119909lowast 119909lowast 119902)
(3)
Case 2 For all 119909 isin 119874(119909lowast 120575) cap (119883 119878) there exists at least oneindex 119894
0isin 119868 such that 119892
1198940(119909) ge 0 it follows that
min (0max (119891 (119909) minus 119891 (119909lowast) 119892119894(119909) 119894 isin 119868)) = 0 (4)
Thus
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
lt minus arg 1 = 119875 (119909lowast 119909lowast 119902)
(5)
The above discussion indicates that 119909lowast is a strictly localmaximizer of 119875(119909 119909lowast 119902)
Theorem 3 Assuming that 119909lowast isin 119871(119875) then for any 119909 isin 1198781
119909 = 119909lowast or 119909 isin 119883119878 it holds that 0 notin 120597119875(119909 119909lowast1 119902) when 119902 gt 0
is big enough
Mathematical Problems in Engineering 3
Proof For any 119909 isin 1198781 119909 = 119909lowast or 119909 isin 119883 119878 we have
min(0max(119891(119909) minus 119891(119909lowast) 119892119894(119909) 119894 isin 119868)) = 0 and so
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
(6)
Thus
120597119875 (119909 119909lowast 119902)
sub minus1
119902[119891 (119909) minus 119891 (119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
times (120597119891 (119909) +119898
sum119894=1
120582119894120597119892119894(119909))
minus2 (119909 minus 119909lowast)
1 + (1 + 119909 minus 119909lowast2)2
(7)
where 0 le 120582119894le 1 119894 isin 119868Therefore when 119902 gt 0 is big enough
it holds that
⟨120597119875 (119909 119909lowast 119902) 119909 minus 119909lowast
119909 minus 119909lowast⟩
le1
119902[119871119863 +
119898
sum119894=1
max119909isin119883
1003817100381710038171003817119892119894 (119909)1003817100381710038171003817][119871
119891+119898
sum119894=1
119871119892119894]
minus21003817100381710038171003817119909 minus 119909lowast
1003817100381710038171003817
1 + (1 + 119909 minus 119909lowast2)2
lt 0
(8)
which implies that 0 notin 120597119875(119909 119909lowast 119902)
Theorem 4 Assume that 119909lowast isin 119871(119875) but 119909lowast notin 119866(119875) and119888119897 int 119878 = 119888119897119878 If 119902 gt 0 is suitably large then there exists a point1199090isin 119878 such that 119909
0is a local minimizer of 119875(119909 119909lowast 119902)
Proof By the conditions there exists an 1199092isin int 119878 such that
119891(1199092) lt 119891(119909lowast) 119892
119894(1199092) lt 0 119894 isin 119868
Then for any 119909 isin 120597119878
119875 (119909 119909lowast 119902) = minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
119875 (1199092 119909lowast 119902) = minus
1
119902[119891(1199092) minus 119891(119909lowast)]
2
minus arg (1 +10038171003817100381710038171199092 minus 119909lowast
10038171003817100381710038172)
+ 119902[max(119891(1199092) minus 119891(119909lowast) 119892
119894(1199092) 119894 isin 119868)]
3
(9)
Since as 119902 rarr +infin
119875 (119909 119909lowast 119902) ge minus120587
2 119875 (1199092 119909lowast 119902) 997888rarr minusinfin (10)
then when 119902 gt 0 is suitably big we have
119875 (1199092 119909lowast 119902) lt 119875 (119909 119909lowast 119902) forall119909 isin 120597119878 (11)
Assume that the function 119875(119909 119909lowast 119902) reaches its global mini-mizer over 119878 at 119909
0 Since 119878120597119878 is an open set then 119909
0isin 119878120597119878
and it holds that
min119909isin119878
119875 (119909 119909lowast 119902) = min119909isin119878120597119878
119875 (119909 119909lowast 119902)
= 119875 (1199090 119909lowast 119902) le 119875 (119909
2 119909lowast 119902)
(12)
In the following we will prove that
119891 (1199090) lt 119891 (119909lowast) 119892
119894(1199090) lt 0 119894 isin 119868 (13)
which leads to the resultThe proof is by contradiction Suppose that
119891 (1199090) ge 119891 (119909lowast) 119892
119894(1199090) lt 0 119894 isin 119868 (14)
Then as 119902 rarr +infin
119875 (1199090 119909lowast 119902) 997888rarr minus arg (1 +
10038171003817100381710038171199090 minus 119909lowast10038171003817100381710038172) gt minusinfin (15)
which implies that when 119902 gt 0 is suitably big we have
119875 (1199090 119909lowast 119902) gt 119875 (119909
2 119909lowast 119902) (16)
This is a contradiction
3 Algorithm and Numerical Examples
Based on the properties of the proposed filled function wenowgive a corresponding filled function algorithmas follows
31 Filled Function Algorithm FFAM
Initialization Step
(1) Set a disturbance constant 120572 = 01(2) Select an upper bound of 119902denoted by 119902
119906and set 119902
119906=
108(3) Select directions 119890
119896 119896 = 1 2 119897 with integer 119897 =
2119899 where 119899 is the number of variables(4) Set 119896 = 1
Main Step
(1) Start from an initial point 119909 minimize the problem(119875) by implementing a nonsmooth local search pro-cedure and obtain the first localminimizer119909lowast
1of119891(119909)
(2) Let 119902 = 1
4 Mathematical Problems in Engineering
(3) Construct a filled function at 119909lowast1
119875 (119909 119909lowast1 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast
1) +119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
1
10038171003817100381710038172)
+ 119902[min (0max (119891 (119909) minus 119891 (119909lowast1) 119892119894(119909) 119894 isin 119868))]
3
(17)
(4) If 119896 gt 119897 then go to (7) Else set 119909 = 119909lowast1
+ 120572119890119896as an
initial point minimize 119875(119909 119909lowast 119902) by implementing anonsmooth local search algorithm and obtain a localminimizer 119909
119896
(5) If 119909119896isin 119878 then set 119896 = 119896 + 1 and go to (4) Else go to
(6)(6) If 119909
119896meets 119891(119909
119896) lt 119891(119909lowast
1) then set 119909 = 119909
119896and
119896 = 1 Start from 119909 as a new initial point minimizethe problem (119875) by using a local search algorithmand obtain another local minimizer 119909lowast
2of 119891(119909) with
119891(119909lowast2) lt 119891(119909lowast
1) Set 119909lowast
1= 119909lowast2and go to (2) Else go to
(7)(7) Increase 119902 by setting 119902 = 10119902(8) If 119902 le 119902
119906 then set 119896 = 1 and go to (3) Else the
algorithm stops and 119909lowast1is taken as a global minimizer
of the problem (119875)
At the end of this section we make a few remarks below(1) Algorithm FFAM is comprised of two stages local
minimization and filling In stage 1 a local minimizer 119909lowast1
of 119891(119909) is identified In stage 2 filled function 119875(119909 119909lowast1 119902) is
constructed and then minimized Stage 2 terminates whenone point 119909
119896isin 1198782is located Then algorithm FFAM
reenters into stage 1 with 119909119896as an initial point to find a new
minimizer 119909lowast2of 119891(119909) (if such one minimizer exists) and so
onThe above process is repeated until some certain specifiedstopping criteria are met and then the last local minimizer isregarded as a global minimizer
(2)Themotivation andmechanism behind the algorithmFFAM are given below
In Step (3) of the Initialization Step we can choosedirections 119890
119896 119896 = 1 2 119897 as positive and negative unit
coordinate vectors For example when 119899 = 2 the directionscan be chosen as (1 0) (0 1) (minus1 0) and (0 minus1)
In Step (1) and Step (6) of the Main Step we can obtain alocal optimizer of the problem (119875) by using any nonsmoothconstrained local optimization procedure such as Bundlemethods and Powellrsquos method In Step (4) of the MainStep we can minimize the proposed filled function by usingHybrid Hooke and Jeeves-Direct Method for NonsmoothOptimization [10] Mesh Adaptive Direct Search Algorithmsfor Constrained Optimization [11] and so forth
(3) The proposed filled function algorithm can also beapplied to smooth constrained optimization Any smoothlocal minimization procedure in the minimization phase
Table 1 Computational results with initial point (minus1 minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1 minus1) (minus19802 minus00132 minus00006) minus194102 (11931 06332 minus11932) (19889 minus00001 minus00111) minus59477
can be used such as conjugate gradient method and quasi-Newton method
32 Numerical Examples Weperform the numerical tests forthree examples All the numerical tests are programmed inFortran 95 In nonsmooth case we search for the local mini-mizers by using Hybrid Hooke and Jeeves-Direct Method forNonsmooth Optimization [10] and theMesh Adaptive DirectSearch Algorithms for Constrained Optimization [11] Insmooth case we use penalty function method and conjugategradient method to get the local minimizers
The following are the three examples and their numericalresults And the symbols used in the tables are explainedbelow
119896 the iteration number in finding the 119896th localminimizer119909119896 the 119896th new initial point in finding the 119896th local
minimizer119909lowast119896 the 119896th local minimizer
119891(119909lowast119896) the function value of the 119896th local minimizer
Problem 1 Consider
min 119891 (119909) = minus11990921+ 11990922+ 11990923minus 1199091
st 11990921+ 11990922+ 11990923minus 4 le 0
min 1199092minus 1199093 1199093 le 0
(18)
Algorithm FFAM succeeds in finding an approximate globalminimizer 119909lowast = (19889 minus00001 minus00111)119879 with 119891(119909lowast) =minus59477 The computational results are given in Table 1
Problem 2 Consider
min 119891 (119909) = minus20 exp(minus02radic10038161003816100381610038161199091
1003816100381610038161003816 +10038161003816100381610038161199092
10038161003816100381610038162
)
minus exp(cos (2120587119909
1) + cos (2120587119909
2)
2) + 20
st 11990921+ 11990922le 300
21199091+ 1199092le 4
minus 30 le 119909119894le 30 119894 = 1 2
(19)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (0 0)119879 with 119891(119909lowast) = minus27183 The computationalresults are listed in Table 2
Mathematical Problems in Engineering 5
Table 2 Computational results with initial point (minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1) (minus150000 00000) 571642 (minus10585 05165) (00001 minus02094) minus036903 (00007 minus00435) (00000 00000) minus27183
Table 3 Computational results with initial point (0 0)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (0 0) (06116 34423) minus405412 (21653 22546) (23295 31780) minus55081
Problem 3 Consider
min 119891 (119909) = minus1199091minus 1199092
st 1199092le 211990941minus 811990931+ 811990921+ 2
1199092le 411990941minus 321199093
1+ 881199092
1minus 96119909
1+ 36
0 le 1199091le 3
0 le 1199092le 4
(20)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (23295 31780)119879 with 119891(119909lowast) = minus55081 This problemis taken from [12] We give this numerical example here toillustrate that algorithm FFAM is also suitable for smoothconstrained global optimization The computational resultsare given in Table 3
4 Conclusion
In this paper we present a new filled function for bothnonsmooth and smooth constrained global optimizationand investigate its properties The filled function containsonly one parameter which can be readily adjusted in theprocess of minimization We also design a correspondingfilled function algorithm Moreover in order to demonstratethe performance of the proposed filled function method wemake three numerical tests The preliminary computationalresults show that the proposed filled function approach ispromising
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thisworkwas supported by theNNSF of China (nos 11471102and 11001248) the SEDF under Grant no 12YZ178 theKey Discipline ldquoApplied Mathematicsrdquo of SSPU under noA30XK1322100 and NNSF of Zhejiang (no LY13A010006)
References
[1] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization Volume 22 Heuristic Approaches Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization vol 62 of Nonconvex Optimization and Its Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 2002
[3] R Horst and P M Pardalos Eds Handbook of Global Opti-mization vol 2 ofNonconvex Optimization and Its ApplicationsKluwer Academic Publishers Dordrecht The Netherlands1995
[4] R P Ge and Y F Qin ldquoA class of filled functions for findingglobal minimizers of a function of several variablesrdquo Journal ofOptimization Theory and Applications vol 54 no 2 pp 241ndash252 1987
[5] W Wang Y Yang and L Zhang ldquoUnification of filled functionand tunnelling function in global optimizationrdquo Acta Mathe-maticaeApplicatae Sinica English Series vol 23 no 1 pp 59ndash662007
[6] Z Xu H Huang P M Pardalos and C Xu ldquoFilled functionsfor unconstrained global optimizationrdquo Journal of Global Opti-mization vol 20 no 1 pp 49ndash65 2001
[7] Y-J Yang and Y-L Shang ldquoA new filled function method forunconstrained global optimizationrdquo Applied Mathematics andComputation vol 173 no 1 pp 501ndash512 2006
[8] A V Levy and A Montalvo ldquoThe tunneling algorithm for theglobal minimization of functionsrdquo SIAM Journal on Scientificand Statistical Computing vol 6 no 1 pp 15ndash29 1985
[9] Y Wang and J Zhang ldquoA new constructing auxiliary functionmethod for global optimizationrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1396ndash1410 2008
[10] C J Price B L Robertson and M Reale ldquoA hybrid Hookeand Jeevesmdashdirect method for non-smooth optimizationrdquoAdvanced Modeling and Optimization vol 11 no 1 pp 43ndash612009
[11] C Audet and J Dennis ldquoMesh adaptive direct search algorithmsfor constrained optimizationrdquo SIAM Journal on Optimizationvol 17 no 1 pp 188ndash217 2006
[12] C A Floudas and P M Pardalos A Collection of Test Prob-lems for Constrained Global Optimization Algorithms SpringerBerlin Germany 1990
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Mathematical Problems in Engineering
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Proof For any 119909 isin 1198781 119909 = 119909lowast or 119909 isin 119883 119878 we have
min(0max(119891(119909) minus 119891(119909lowast) 119892119894(119909) 119894 isin 119868)) = 0 and so
119875 (119909 119909lowast 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
(6)
Thus
120597119875 (119909 119909lowast 119902)
sub minus1
119902[119891 (119909) minus 119891 (119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
times (120597119891 (119909) +119898
sum119894=1
120582119894120597119892119894(119909))
minus2 (119909 minus 119909lowast)
1 + (1 + 119909 minus 119909lowast2)2
(7)
where 0 le 120582119894le 1 119894 isin 119868Therefore when 119902 gt 0 is big enough
it holds that
⟨120597119875 (119909 119909lowast 119902) 119909 minus 119909lowast
119909 minus 119909lowast⟩
le1
119902[119871119863 +
119898
sum119894=1
max119909isin119883
1003817100381710038171003817119892119894 (119909)1003817100381710038171003817][119871
119891+119898
sum119894=1
119871119892119894]
minus21003817100381710038171003817119909 minus 119909lowast
1003817100381710038171003817
1 + (1 + 119909 minus 119909lowast2)2
lt 0
(8)
which implies that 0 notin 120597119875(119909 119909lowast 119902)
Theorem 4 Assume that 119909lowast isin 119871(119875) but 119909lowast notin 119866(119875) and119888119897 int 119878 = 119888119897119878 If 119902 gt 0 is suitably large then there exists a point1199090isin 119878 such that 119909
0is a local minimizer of 119875(119909 119909lowast 119902)
Proof By the conditions there exists an 1199092isin int 119878 such that
119891(1199092) lt 119891(119909lowast) 119892
119894(1199092) lt 0 119894 isin 119868
Then for any 119909 isin 120597119878
119875 (119909 119909lowast 119902) = minus1
119902[119891(119909) minus 119891(119909lowast) +
119898
sum119894=1
max (0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
10038171003817100381710038172)
119875 (1199092 119909lowast 119902) = minus
1
119902[119891(1199092) minus 119891(119909lowast)]
2
minus arg (1 +10038171003817100381710038171199092 minus 119909lowast
10038171003817100381710038172)
+ 119902[max(119891(1199092) minus 119891(119909lowast) 119892
119894(1199092) 119894 isin 119868)]
3
(9)
Since as 119902 rarr +infin
119875 (119909 119909lowast 119902) ge minus120587
2 119875 (1199092 119909lowast 119902) 997888rarr minusinfin (10)
then when 119902 gt 0 is suitably big we have
119875 (1199092 119909lowast 119902) lt 119875 (119909 119909lowast 119902) forall119909 isin 120597119878 (11)
Assume that the function 119875(119909 119909lowast 119902) reaches its global mini-mizer over 119878 at 119909
0 Since 119878120597119878 is an open set then 119909
0isin 119878120597119878
and it holds that
min119909isin119878
119875 (119909 119909lowast 119902) = min119909isin119878120597119878
119875 (119909 119909lowast 119902)
= 119875 (1199090 119909lowast 119902) le 119875 (119909
2 119909lowast 119902)
(12)
In the following we will prove that
119891 (1199090) lt 119891 (119909lowast) 119892
119894(1199090) lt 0 119894 isin 119868 (13)
which leads to the resultThe proof is by contradiction Suppose that
119891 (1199090) ge 119891 (119909lowast) 119892
119894(1199090) lt 0 119894 isin 119868 (14)
Then as 119902 rarr +infin
119875 (1199090 119909lowast 119902) 997888rarr minus arg (1 +
10038171003817100381710038171199090 minus 119909lowast10038171003817100381710038172) gt minusinfin (15)
which implies that when 119902 gt 0 is suitably big we have
119875 (1199090 119909lowast 119902) gt 119875 (119909
2 119909lowast 119902) (16)
This is a contradiction
3 Algorithm and Numerical Examples
Based on the properties of the proposed filled function wenowgive a corresponding filled function algorithmas follows
31 Filled Function Algorithm FFAM
Initialization Step
(1) Set a disturbance constant 120572 = 01(2) Select an upper bound of 119902denoted by 119902
119906and set 119902
119906=
108(3) Select directions 119890
119896 119896 = 1 2 119897 with integer 119897 =
2119899 where 119899 is the number of variables(4) Set 119896 = 1
Main Step
(1) Start from an initial point 119909 minimize the problem(119875) by implementing a nonsmooth local search pro-cedure and obtain the first localminimizer119909lowast
1of119891(119909)
(2) Let 119902 = 1
4 Mathematical Problems in Engineering
(3) Construct a filled function at 119909lowast1
119875 (119909 119909lowast1 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast
1) +119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
1
10038171003817100381710038172)
+ 119902[min (0max (119891 (119909) minus 119891 (119909lowast1) 119892119894(119909) 119894 isin 119868))]
3
(17)
(4) If 119896 gt 119897 then go to (7) Else set 119909 = 119909lowast1
+ 120572119890119896as an
initial point minimize 119875(119909 119909lowast 119902) by implementing anonsmooth local search algorithm and obtain a localminimizer 119909
119896
(5) If 119909119896isin 119878 then set 119896 = 119896 + 1 and go to (4) Else go to
(6)(6) If 119909
119896meets 119891(119909
119896) lt 119891(119909lowast
1) then set 119909 = 119909
119896and
119896 = 1 Start from 119909 as a new initial point minimizethe problem (119875) by using a local search algorithmand obtain another local minimizer 119909lowast
2of 119891(119909) with
119891(119909lowast2) lt 119891(119909lowast
1) Set 119909lowast
1= 119909lowast2and go to (2) Else go to
(7)(7) Increase 119902 by setting 119902 = 10119902(8) If 119902 le 119902
119906 then set 119896 = 1 and go to (3) Else the
algorithm stops and 119909lowast1is taken as a global minimizer
of the problem (119875)
At the end of this section we make a few remarks below(1) Algorithm FFAM is comprised of two stages local
minimization and filling In stage 1 a local minimizer 119909lowast1
of 119891(119909) is identified In stage 2 filled function 119875(119909 119909lowast1 119902) is
constructed and then minimized Stage 2 terminates whenone point 119909
119896isin 1198782is located Then algorithm FFAM
reenters into stage 1 with 119909119896as an initial point to find a new
minimizer 119909lowast2of 119891(119909) (if such one minimizer exists) and so
onThe above process is repeated until some certain specifiedstopping criteria are met and then the last local minimizer isregarded as a global minimizer
(2)Themotivation andmechanism behind the algorithmFFAM are given below
In Step (3) of the Initialization Step we can choosedirections 119890
119896 119896 = 1 2 119897 as positive and negative unit
coordinate vectors For example when 119899 = 2 the directionscan be chosen as (1 0) (0 1) (minus1 0) and (0 minus1)
In Step (1) and Step (6) of the Main Step we can obtain alocal optimizer of the problem (119875) by using any nonsmoothconstrained local optimization procedure such as Bundlemethods and Powellrsquos method In Step (4) of the MainStep we can minimize the proposed filled function by usingHybrid Hooke and Jeeves-Direct Method for NonsmoothOptimization [10] Mesh Adaptive Direct Search Algorithmsfor Constrained Optimization [11] and so forth
(3) The proposed filled function algorithm can also beapplied to smooth constrained optimization Any smoothlocal minimization procedure in the minimization phase
Table 1 Computational results with initial point (minus1 minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1 minus1) (minus19802 minus00132 minus00006) minus194102 (11931 06332 minus11932) (19889 minus00001 minus00111) minus59477
can be used such as conjugate gradient method and quasi-Newton method
32 Numerical Examples Weperform the numerical tests forthree examples All the numerical tests are programmed inFortran 95 In nonsmooth case we search for the local mini-mizers by using Hybrid Hooke and Jeeves-Direct Method forNonsmooth Optimization [10] and theMesh Adaptive DirectSearch Algorithms for Constrained Optimization [11] Insmooth case we use penalty function method and conjugategradient method to get the local minimizers
The following are the three examples and their numericalresults And the symbols used in the tables are explainedbelow
119896 the iteration number in finding the 119896th localminimizer119909119896 the 119896th new initial point in finding the 119896th local
minimizer119909lowast119896 the 119896th local minimizer
119891(119909lowast119896) the function value of the 119896th local minimizer
Problem 1 Consider
min 119891 (119909) = minus11990921+ 11990922+ 11990923minus 1199091
st 11990921+ 11990922+ 11990923minus 4 le 0
min 1199092minus 1199093 1199093 le 0
(18)
Algorithm FFAM succeeds in finding an approximate globalminimizer 119909lowast = (19889 minus00001 minus00111)119879 with 119891(119909lowast) =minus59477 The computational results are given in Table 1
Problem 2 Consider
min 119891 (119909) = minus20 exp(minus02radic10038161003816100381610038161199091
1003816100381610038161003816 +10038161003816100381610038161199092
10038161003816100381610038162
)
minus exp(cos (2120587119909
1) + cos (2120587119909
2)
2) + 20
st 11990921+ 11990922le 300
21199091+ 1199092le 4
minus 30 le 119909119894le 30 119894 = 1 2
(19)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (0 0)119879 with 119891(119909lowast) = minus27183 The computationalresults are listed in Table 2
Mathematical Problems in Engineering 5
Table 2 Computational results with initial point (minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1) (minus150000 00000) 571642 (minus10585 05165) (00001 minus02094) minus036903 (00007 minus00435) (00000 00000) minus27183
Table 3 Computational results with initial point (0 0)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (0 0) (06116 34423) minus405412 (21653 22546) (23295 31780) minus55081
Problem 3 Consider
min 119891 (119909) = minus1199091minus 1199092
st 1199092le 211990941minus 811990931+ 811990921+ 2
1199092le 411990941minus 321199093
1+ 881199092
1minus 96119909
1+ 36
0 le 1199091le 3
0 le 1199092le 4
(20)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (23295 31780)119879 with 119891(119909lowast) = minus55081 This problemis taken from [12] We give this numerical example here toillustrate that algorithm FFAM is also suitable for smoothconstrained global optimization The computational resultsare given in Table 3
4 Conclusion
In this paper we present a new filled function for bothnonsmooth and smooth constrained global optimizationand investigate its properties The filled function containsonly one parameter which can be readily adjusted in theprocess of minimization We also design a correspondingfilled function algorithm Moreover in order to demonstratethe performance of the proposed filled function method wemake three numerical tests The preliminary computationalresults show that the proposed filled function approach ispromising
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thisworkwas supported by theNNSF of China (nos 11471102and 11001248) the SEDF under Grant no 12YZ178 theKey Discipline ldquoApplied Mathematicsrdquo of SSPU under noA30XK1322100 and NNSF of Zhejiang (no LY13A010006)
References
[1] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization Volume 22 Heuristic Approaches Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization vol 62 of Nonconvex Optimization and Its Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 2002
[3] R Horst and P M Pardalos Eds Handbook of Global Opti-mization vol 2 ofNonconvex Optimization and Its ApplicationsKluwer Academic Publishers Dordrecht The Netherlands1995
[4] R P Ge and Y F Qin ldquoA class of filled functions for findingglobal minimizers of a function of several variablesrdquo Journal ofOptimization Theory and Applications vol 54 no 2 pp 241ndash252 1987
[5] W Wang Y Yang and L Zhang ldquoUnification of filled functionand tunnelling function in global optimizationrdquo Acta Mathe-maticaeApplicatae Sinica English Series vol 23 no 1 pp 59ndash662007
[6] Z Xu H Huang P M Pardalos and C Xu ldquoFilled functionsfor unconstrained global optimizationrdquo Journal of Global Opti-mization vol 20 no 1 pp 49ndash65 2001
[7] Y-J Yang and Y-L Shang ldquoA new filled function method forunconstrained global optimizationrdquo Applied Mathematics andComputation vol 173 no 1 pp 501ndash512 2006
[8] A V Levy and A Montalvo ldquoThe tunneling algorithm for theglobal minimization of functionsrdquo SIAM Journal on Scientificand Statistical Computing vol 6 no 1 pp 15ndash29 1985
[9] Y Wang and J Zhang ldquoA new constructing auxiliary functionmethod for global optimizationrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1396ndash1410 2008
[10] C J Price B L Robertson and M Reale ldquoA hybrid Hookeand Jeevesmdashdirect method for non-smooth optimizationrdquoAdvanced Modeling and Optimization vol 11 no 1 pp 43ndash612009
[11] C Audet and J Dennis ldquoMesh adaptive direct search algorithmsfor constrained optimizationrdquo SIAM Journal on Optimizationvol 17 no 1 pp 188ndash217 2006
[12] C A Floudas and P M Pardalos A Collection of Test Prob-lems for Constrained Global Optimization Algorithms SpringerBerlin Germany 1990
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
(3) Construct a filled function at 119909lowast1
119875 (119909 119909lowast1 119902)
= minus1
119902[119891(119909) minus 119891(119909lowast
1) +119898
sum119894=1
max(0 119892119894(119909))]
2
minus arg (1 +1003817100381710038171003817119909 minus 119909lowast
1
10038171003817100381710038172)
+ 119902[min (0max (119891 (119909) minus 119891 (119909lowast1) 119892119894(119909) 119894 isin 119868))]
3
(17)
(4) If 119896 gt 119897 then go to (7) Else set 119909 = 119909lowast1
+ 120572119890119896as an
initial point minimize 119875(119909 119909lowast 119902) by implementing anonsmooth local search algorithm and obtain a localminimizer 119909
119896
(5) If 119909119896isin 119878 then set 119896 = 119896 + 1 and go to (4) Else go to
(6)(6) If 119909
119896meets 119891(119909
119896) lt 119891(119909lowast
1) then set 119909 = 119909
119896and
119896 = 1 Start from 119909 as a new initial point minimizethe problem (119875) by using a local search algorithmand obtain another local minimizer 119909lowast
2of 119891(119909) with
119891(119909lowast2) lt 119891(119909lowast
1) Set 119909lowast
1= 119909lowast2and go to (2) Else go to
(7)(7) Increase 119902 by setting 119902 = 10119902(8) If 119902 le 119902
119906 then set 119896 = 1 and go to (3) Else the
algorithm stops and 119909lowast1is taken as a global minimizer
of the problem (119875)
At the end of this section we make a few remarks below(1) Algorithm FFAM is comprised of two stages local
minimization and filling In stage 1 a local minimizer 119909lowast1
of 119891(119909) is identified In stage 2 filled function 119875(119909 119909lowast1 119902) is
constructed and then minimized Stage 2 terminates whenone point 119909
119896isin 1198782is located Then algorithm FFAM
reenters into stage 1 with 119909119896as an initial point to find a new
minimizer 119909lowast2of 119891(119909) (if such one minimizer exists) and so
onThe above process is repeated until some certain specifiedstopping criteria are met and then the last local minimizer isregarded as a global minimizer
(2)Themotivation andmechanism behind the algorithmFFAM are given below
In Step (3) of the Initialization Step we can choosedirections 119890
119896 119896 = 1 2 119897 as positive and negative unit
coordinate vectors For example when 119899 = 2 the directionscan be chosen as (1 0) (0 1) (minus1 0) and (0 minus1)
In Step (1) and Step (6) of the Main Step we can obtain alocal optimizer of the problem (119875) by using any nonsmoothconstrained local optimization procedure such as Bundlemethods and Powellrsquos method In Step (4) of the MainStep we can minimize the proposed filled function by usingHybrid Hooke and Jeeves-Direct Method for NonsmoothOptimization [10] Mesh Adaptive Direct Search Algorithmsfor Constrained Optimization [11] and so forth
(3) The proposed filled function algorithm can also beapplied to smooth constrained optimization Any smoothlocal minimization procedure in the minimization phase
Table 1 Computational results with initial point (minus1 minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1 minus1) (minus19802 minus00132 minus00006) minus194102 (11931 06332 minus11932) (19889 minus00001 minus00111) minus59477
can be used such as conjugate gradient method and quasi-Newton method
32 Numerical Examples Weperform the numerical tests forthree examples All the numerical tests are programmed inFortran 95 In nonsmooth case we search for the local mini-mizers by using Hybrid Hooke and Jeeves-Direct Method forNonsmooth Optimization [10] and theMesh Adaptive DirectSearch Algorithms for Constrained Optimization [11] Insmooth case we use penalty function method and conjugategradient method to get the local minimizers
The following are the three examples and their numericalresults And the symbols used in the tables are explainedbelow
119896 the iteration number in finding the 119896th localminimizer119909119896 the 119896th new initial point in finding the 119896th local
minimizer119909lowast119896 the 119896th local minimizer
119891(119909lowast119896) the function value of the 119896th local minimizer
Problem 1 Consider
min 119891 (119909) = minus11990921+ 11990922+ 11990923minus 1199091
st 11990921+ 11990922+ 11990923minus 4 le 0
min 1199092minus 1199093 1199093 le 0
(18)
Algorithm FFAM succeeds in finding an approximate globalminimizer 119909lowast = (19889 minus00001 minus00111)119879 with 119891(119909lowast) =minus59477 The computational results are given in Table 1
Problem 2 Consider
min 119891 (119909) = minus20 exp(minus02radic10038161003816100381610038161199091
1003816100381610038161003816 +10038161003816100381610038161199092
10038161003816100381610038162
)
minus exp(cos (2120587119909
1) + cos (2120587119909
2)
2) + 20
st 11990921+ 11990922le 300
21199091+ 1199092le 4
minus 30 le 119909119894le 30 119894 = 1 2
(19)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (0 0)119879 with 119891(119909lowast) = minus27183 The computationalresults are listed in Table 2
Mathematical Problems in Engineering 5
Table 2 Computational results with initial point (minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1) (minus150000 00000) 571642 (minus10585 05165) (00001 minus02094) minus036903 (00007 minus00435) (00000 00000) minus27183
Table 3 Computational results with initial point (0 0)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (0 0) (06116 34423) minus405412 (21653 22546) (23295 31780) minus55081
Problem 3 Consider
min 119891 (119909) = minus1199091minus 1199092
st 1199092le 211990941minus 811990931+ 811990921+ 2
1199092le 411990941minus 321199093
1+ 881199092
1minus 96119909
1+ 36
0 le 1199091le 3
0 le 1199092le 4
(20)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (23295 31780)119879 with 119891(119909lowast) = minus55081 This problemis taken from [12] We give this numerical example here toillustrate that algorithm FFAM is also suitable for smoothconstrained global optimization The computational resultsare given in Table 3
4 Conclusion
In this paper we present a new filled function for bothnonsmooth and smooth constrained global optimizationand investigate its properties The filled function containsonly one parameter which can be readily adjusted in theprocess of minimization We also design a correspondingfilled function algorithm Moreover in order to demonstratethe performance of the proposed filled function method wemake three numerical tests The preliminary computationalresults show that the proposed filled function approach ispromising
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thisworkwas supported by theNNSF of China (nos 11471102and 11001248) the SEDF under Grant no 12YZ178 theKey Discipline ldquoApplied Mathematicsrdquo of SSPU under noA30XK1322100 and NNSF of Zhejiang (no LY13A010006)
References
[1] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization Volume 22 Heuristic Approaches Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization vol 62 of Nonconvex Optimization and Its Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 2002
[3] R Horst and P M Pardalos Eds Handbook of Global Opti-mization vol 2 ofNonconvex Optimization and Its ApplicationsKluwer Academic Publishers Dordrecht The Netherlands1995
[4] R P Ge and Y F Qin ldquoA class of filled functions for findingglobal minimizers of a function of several variablesrdquo Journal ofOptimization Theory and Applications vol 54 no 2 pp 241ndash252 1987
[5] W Wang Y Yang and L Zhang ldquoUnification of filled functionand tunnelling function in global optimizationrdquo Acta Mathe-maticaeApplicatae Sinica English Series vol 23 no 1 pp 59ndash662007
[6] Z Xu H Huang P M Pardalos and C Xu ldquoFilled functionsfor unconstrained global optimizationrdquo Journal of Global Opti-mization vol 20 no 1 pp 49ndash65 2001
[7] Y-J Yang and Y-L Shang ldquoA new filled function method forunconstrained global optimizationrdquo Applied Mathematics andComputation vol 173 no 1 pp 501ndash512 2006
[8] A V Levy and A Montalvo ldquoThe tunneling algorithm for theglobal minimization of functionsrdquo SIAM Journal on Scientificand Statistical Computing vol 6 no 1 pp 15ndash29 1985
[9] Y Wang and J Zhang ldquoA new constructing auxiliary functionmethod for global optimizationrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1396ndash1410 2008
[10] C J Price B L Robertson and M Reale ldquoA hybrid Hookeand Jeevesmdashdirect method for non-smooth optimizationrdquoAdvanced Modeling and Optimization vol 11 no 1 pp 43ndash612009
[11] C Audet and J Dennis ldquoMesh adaptive direct search algorithmsfor constrained optimizationrdquo SIAM Journal on Optimizationvol 17 no 1 pp 188ndash217 2006
[12] C A Floudas and P M Pardalos A Collection of Test Prob-lems for Constrained Global Optimization Algorithms SpringerBerlin Germany 1990
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Table 2 Computational results with initial point (minus1 minus1)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (minus1 minus1) (minus150000 00000) 571642 (minus10585 05165) (00001 minus02094) minus036903 (00007 minus00435) (00000 00000) minus27183
Table 3 Computational results with initial point (0 0)
119896 119909119896
119909lowast119896
119891(119909lowast119896)
1 (0 0) (06116 34423) minus405412 (21653 22546) (23295 31780) minus55081
Problem 3 Consider
min 119891 (119909) = minus1199091minus 1199092
st 1199092le 211990941minus 811990931+ 811990921+ 2
1199092le 411990941minus 321199093
1+ 881199092
1minus 96119909
1+ 36
0 le 1199091le 3
0 le 1199092le 4
(20)
Algorithm FFAM succeeds in finding a global minimizer119909lowast = (23295 31780)119879 with 119891(119909lowast) = minus55081 This problemis taken from [12] We give this numerical example here toillustrate that algorithm FFAM is also suitable for smoothconstrained global optimization The computational resultsare given in Table 3
4 Conclusion
In this paper we present a new filled function for bothnonsmooth and smooth constrained global optimizationand investigate its properties The filled function containsonly one parameter which can be readily adjusted in theprocess of minimization We also design a correspondingfilled function algorithm Moreover in order to demonstratethe performance of the proposed filled function method wemake three numerical tests The preliminary computationalresults show that the proposed filled function approach ispromising
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thisworkwas supported by theNNSF of China (nos 11471102and 11001248) the SEDF under Grant no 12YZ178 theKey Discipline ldquoApplied Mathematicsrdquo of SSPU under noA30XK1322100 and NNSF of Zhejiang (no LY13A010006)
References
[1] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization Volume 22 Heuristic Approaches Kluwer Aca-demic Publishers Dordrecht The Netherlands 2002
[2] P M Pardalos and H E Romeijn Eds Handbook of GlobalOptimization vol 62 of Nonconvex Optimization and Its Appli-cations Kluwer Academic Publishers Dordrecht The Nether-lands 2002
[3] R Horst and P M Pardalos Eds Handbook of Global Opti-mization vol 2 ofNonconvex Optimization and Its ApplicationsKluwer Academic Publishers Dordrecht The Netherlands1995
[4] R P Ge and Y F Qin ldquoA class of filled functions for findingglobal minimizers of a function of several variablesrdquo Journal ofOptimization Theory and Applications vol 54 no 2 pp 241ndash252 1987
[5] W Wang Y Yang and L Zhang ldquoUnification of filled functionand tunnelling function in global optimizationrdquo Acta Mathe-maticaeApplicatae Sinica English Series vol 23 no 1 pp 59ndash662007
[6] Z Xu H Huang P M Pardalos and C Xu ldquoFilled functionsfor unconstrained global optimizationrdquo Journal of Global Opti-mization vol 20 no 1 pp 49ndash65 2001
[7] Y-J Yang and Y-L Shang ldquoA new filled function method forunconstrained global optimizationrdquo Applied Mathematics andComputation vol 173 no 1 pp 501ndash512 2006
[8] A V Levy and A Montalvo ldquoThe tunneling algorithm for theglobal minimization of functionsrdquo SIAM Journal on Scientificand Statistical Computing vol 6 no 1 pp 15ndash29 1985
[9] Y Wang and J Zhang ldquoA new constructing auxiliary functionmethod for global optimizationrdquo Mathematical and ComputerModelling vol 47 no 11-12 pp 1396ndash1410 2008
[10] C J Price B L Robertson and M Reale ldquoA hybrid Hookeand Jeevesmdashdirect method for non-smooth optimizationrdquoAdvanced Modeling and Optimization vol 11 no 1 pp 43ndash612009
[11] C Audet and J Dennis ldquoMesh adaptive direct search algorithmsfor constrained optimizationrdquo SIAM Journal on Optimizationvol 17 no 1 pp 188ndash217 2006
[12] C A Floudas and P M Pardalos A Collection of Test Prob-lems for Constrained Global Optimization Algorithms SpringerBerlin Germany 1990
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of