New One-Dimensional Search Iteration Algorithm and

Research ArticleNew One-Dimensional Search Iteration Algorithm andEngineering Application

Yiping Luo 1 Jinhao Meng 1 Defa Wang 2 and Guobin Xue 3

1Institute of Water Resources and Hydro-Electric Engineering Xirsquoan University of Technology Xirsquoan 710048 China2Institute of Civil Engineering Xirsquoan University of Technology Xirsquoan 710048 China3State Grid Gansu Electric Power Company Economic Technology Research Institute Gansu 730050 China

Correspondence should be addressed to Defa Wang 854244066qqcom

Received 7 May 2021 Revised 1 October 2021 Accepted 4 October 2021 Published 2 November 2021

Academic Editor Mahdi Mohammadpour

Copyright copy 2021 Yiping Luo et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In structural optimization design obtaining the optimal solution of the objective function is the key to optimal design and one-dimensional search is one of the important methods for function optimization-e Golden Section method is the main method ofone-dimensional search which has better convergence and stability Based on the solution of the Golden Section method thispaper proposes an efficient one-dimensional search algorithm which has the advantages of fast convergence and good stabilityAn objective function calculation formula is introduced to compare and analyse this method with the Golden Section methodNewton method and Fibonacci method It is concluded that when the accuracy is set to 01 the new algorithm needs 3 iterationsto obtain the target value -e Golden Section method takes 11 iterations and the Fibonacci method requires 11 iterations -eNewtonmethod cannot obtain the target valueWhen the accuracy is set to 001 the number of iterations of the newmethod is stillthe least -e optimized design of the T-section beam is introduced for engineering application research When the accuracy is setto 01 the new method needs 3 iterations to obtain the target value and the Golden Section method requires 13 iterations Whenthe accuracy is set to 001 the new method requires 4 iterations and the Golden Section method requires 18 iterations -e newmethod has significant advantages in the one-dimensional search optimization problem

1 Introduction

Optimization problems often appear in the fields of engi-neering and scientific research Many scholars have devel-oped different optimization methods -is is becausetraditional optimization methods have some inherentshortcomings such as local optimal stagnation and searchspace expansion [1] In order to solve the practical problemsencountered in engineering design and life many scholarshave improved the existing methods Abbasi et al [2]designed an improved Harris hawks (HHO) optimizationalgorithm to optimize the design of tapered roller bearings(TRB) Too Jingwei used HHO optimization algorithmsMEHHO1 and MEHHO2 for optimal feature selection andsolved the problem of feature selection in the classification ofEMG signals [3] Salvia algae group algorithm (SSA) is abionic optimization algorithm established in 2017 based on

the chain search behaviour of Salvia foraging Nautiyal et al[4] proposed an improved SSA algorithm based onGaussian Cauchy and Levy flight which overcomes thetendency of slow convergence and falling into suboptimalsolutions and improves the global search ability Guo et al[5] proposed a coastal ship route planning model based onthe optimized deep Q network (DQN) algorithm to realizethe route planning of ships during navigation Comparedwith the experiment of the traditional algorithm it has betterstability and convergence reducing the calculation timeDeng et al designed an improved quantum evolution al-gorithm (QEA) based on niche coevolution strategy andenhanced particle swarm optimization (PSO) namelyIPOQEA to solve the problem of airport gate allocation Amethod of boarding gate allocation based on IPOQEA isproposed and the effectiveness of the proposed method isverified by the actual operation data of Baiyun Airport [6] In

HindawiShock and VibrationVolume 2021 Article ID 7643555 11 pageshttpsdoiorg10115520217643555

addition Jin et al used mathematical models to optimize thedesign of financial market models and circuit control sys-tems [7ndash10] Seeking the lowest cost design under thepremise of meeting and application is to optimize the design-e design principle is the optimal design the designmethod is the computer and the design method is theoptimal mathematical method [11] Structural optimizationhas been adopted in many structural engineering practicessuch as quilted structures frame structures bridges hy-draulic structures and other optimization designs that havereceived good results Many scholars have conducted a lot ofresearch on the structural design of reinforced concretebeams Liu [12] used the method of structural optimizationto determine the section height of reinforced concrete beamsand changed the traditional iterative calculation method ofdetermining the beam heightso that the beam height wasoptimized to simplify the design and reduce the project costLao [13] considered the load-bearing capacity requirementsand reinforcement structure of the reinforced concrete beamsection to optimize the design of the beam section anddiscussed the application of the 0618 method in the opti-mization design Yu and Li [14] used the improved con-strained nonlinear mixed discrete variable optimizationdesign method (MDOD) to optimize the design of rein-forced concrete single-reinforced beams taking the cost ofbeam unit length as the objective function and the strengthof positive and oblique sections and the structural re-quirements of single-reinforced beams as the constraintconditions

One-dimensional search also known as linear searchrefers to the optimization of a single-variable function and isone of the basic methods for solving unconstrained non-linear programming problems One-dimensional search canbe used independently to solve single-variable optimizationproblems and at the same time it is a common method tosolve multivariable optimization problems Although one-dimensional search itself can be used to solve many one-dimensional optimization problems in science and practiceit is more used as a means of accelerating algorithm con-vergence and is used in conjunction with multidimensionaloptimization methods such as those suitable for solvinglarge-scale unconstrained optimization -e conjugategradient method of the problem is combined with variousquasi-Newton methods to solve numerous multidimen-sional optimization problems As one of the one-dimen-sional search methods the Golden Section method has highaccuracy and does not require high requirements for thedifferentiability and convergence of the function -eGolden Section method has a wide range of applicabilityZhang [15] built a high-precision correlated colour tem-perature calculation method based on a small-scale lookuptable based on the Golden Section method and the resultsshowed that the algorithm has excellent performanceKheldoun et al [16] proposed a new method for tracking themaximum power point of photovoltaic systems based on theGolden Section method optimization technology whichensures that the photovoltaic system can quickly converge tothe maximum power point thereby reducing energy wasteZoubiri et al [17] used the Golden Section method to

optimize the chemical pretreatment of agroindustrial wasteused to extract sugar and the results showed that the GoldenSection algorithm is a useful step to ensure the extraction oftotal sugar Gao et al [18] studied the free vibration char-acteristics of sandwich piezoelectric beams under elasticboundary conditions and thermal environments and in-troduced a search algorithm based on the Golden Sectionmethod search to calculate the required frequency param-eters Finally the correctness and universality of the algo-rithm were verified by simulation software Shi et al [19]applied the Golden Section one-dimensional search methodto unconstrained multivariate optimization problem solv-ing compared it with Newtonrsquos method and dampedNewtonrsquos method and proved that the Golden Section al-gorithm is effective and practical Liu et al [20] used theldquo0618 methodrdquo the Fibonacci method and the parabolamethod to search for the extreme points of the pendulumcompound pendulum period -rough experiments it isfound that the ldquo0618 methodrdquo used 15 times to measure andthe Fibonacci method used 17 times to measure -eFibonacci method has higher search accuracy than theldquoGolden Section methodrdquo but it does not have an advantageof search speed in comparison However some scholarsbelieve that the Golden Section method had shortcomingsQian [21] pointed out that using the Golden Section methodto optimize can only be a unimodal function in the intervaland the scope of application is more limited Zhang andChen [22] believe that the Golden Section method is rep-resented by the division method Although this type ofmethod has global convergence it does not use the prop-erties of the function For some functions with better an-alytical properties the convergence speed of this method istoo slow Zhang et al [23] proposed an improved one-di-mensional search index optimization algorithm -e ex-ample verification results show that the index optimizationmethod is faster than the Golden Section method At thesame time the interval accuracy of the optimal solution ismore accurate and when the solution intervals are similarthe convergence speed is still faster than that of the GoldenSection method Since the Golden Section method shortensthe length of the search interval by 618 each time Liu et al[24] believe that from the programming perspective theGolden Section method only needs to insert one point at atime but the interval shortening rate is fixed at 618-erefore a midpoint method was proposed to increase theshortening rate to 51 -is paper proposes a new one-dimensional search algorithm based on the iterative algo-rithm of the Golden Section method (Algorithm 1)

2 Background Information

-e Golden Section method is one of the main methods foraccurate one-dimensional search By taking test points andcomparing function values the search interval containingthe minimum points is continuously shortened When theinterval length is shortened to a certain extent the points onthe interval all are close to the minimum so each point onthe interval can be regarded as an approximation of theminimum point

2 Shock and Vibration

Theorem 1 Let ψ is a unimodal function on the interval of[ak bk] λk μk isin [ak bk] if f(λk)gtf(μk) then for eachx isin [ak λk] there is ψ(x)geψ(μk) if f(λk)lef(μk) then foreach x isin [μk bk] there is ψ(x)geψ(μk)

Also the shortening rate of each iteration interval is set to0618 so the insertion points λk and μk correspond to

λk ak + 0382 bk minus ak( 1113857

μk ak + 0618 bk minus ak( 1113857(1)

Given the accuracy εgt 0 when the interval length of acertain step |bk minus ak|le ε is reached the iteration is stoppedLet xlowast be the abscissa of the extreme point in the interval andany point within [ak bk] can be taken as the approximatevalue of the abscissa of the extreme point

3 New One-Dimensional Search Algorithm

Based on the Golden Section iterative algorithm this paperproposes a new one-dimensional search algorithm -eprinciple is as follows

Suppose f(x) is a continuous unimodal function in theinterval of [ak bk] and xlowast is the abscissa of minimum pointIf the abscissa of the endpoint of the interval with the smallerfunction value is ak name the endpoint ak and draw a lineparallel to the x-axis through point ak of f(x) this parallelline has an intersection with the function curve on the otherside of the point xlowast Set the abscissa of this intersection as mand call this point m -en draw the intersection functioncurve of the midperpendicular line connecting point ak andpoint m at one point set the abscissa of this intersectionpoint as m2 and call this point m2 Repeat the above op-eration and draw a line parallel to the x-axis through pointm2 of f(x) this parallel line has an intersection with thefunction curve on the other side of the point xlowast Set theabscissa of this intersection as m3 and call this point m3 Itcan be seen that the abscissa of xlowast still falls in the interval[m2 m3] as shown in Figure 1 Compare the absolute valueof |m2 minus m3| with the precision requirement δ If|m2 minus m3|le δ output (m2 minus m3)2 as the abscissa of theminimum point xlowast Otherwise judge the value of m2 andm3

When m2 ltm3 select the iterative insertion points ξk

and ηk and let

ξk m2 + 0382 m3 minus m2( 1113857

ηk m2 + 0618 m3 minus m2( 1113857(2)

Calculate the values of functions f(ξk) and f(ηk)If f(ξk)gtf(ηk) then

ak+1 ξk

bk+1 m3

ξk+1 ηk

ηk+1 ak+1 + 0618 bk+1 minus ak+1( 1113857

(3)

Calculate the values of functions f(ξk+1) and f(ηk+1)Get a new search interval and xlowast still falls within the newinterval ([ak+1 bk+1] ak+1 ξk bk+1 m3)

If f(ξk)ltf(ηk) then

ak+1 m2

bk+1 ηk

ηk+1 ξk

ξk+1 ak+1 + 0382 bk+1 minus ak+1( 1113857

(4)

Calculate the values of functions f(ξk+1) and f(ηk+1)Get a new search interval and xlowast still falls within the newinterval ([ak+1 bk+1] ak+1 m2 bk+1 ηk)

When m3 ltm2 exchange the values of m2 and m3Repeat the above process -e length of the search in-

terval will continue to shrink and approach zero and finallyit will converge to the minimum point xlowast steadily

-e algorithm steps of the new one-dimensional searchalgorithm are as follows

-e first step given the initial search interval [ak bk]accuracy δ and the accuracy requirement δ gt 0 cal-culate the function values f(ak) and f(bk)-e second step if the length of the search interval isless than the accuracy that is |bk minus ak|lt δ then outputxlowast (ak + bk)2 otherwise go to the third step-e third stepWhen the function value f(ak) is less than f(bk) letthe abscissas of point m m2 and m3 be

Given the initial search interval [ak bk] and accuracy requirements εgt 0(1) Let μk ak + 0618(bk minus ak) andψ2 ψ(μk)

(2) Let λk ak + 0382(bk minus ak) andψ1 ψ(λk)

(3) If |bk minus ak|le ε then let xlowast (ak + bk)2 otherwise transfer to (4)(4) If ψ(λk)ltψ(μk) then let bk μk μk λk andψ2 ψ1 go to (2)

If ψ(λk) ψ(μk) ak λk and bk ψ(μk)go to (1)If ψ(λk) ψ(μk) then let ak λk λk μk andψ1 ψ2 go to (5)

(5) Let μk ak + 0618(bk minus ak) [ak bk] go to (3)

ALGORITHM 1 -e Golden Section iterative algorithm

Shock and Vibration 3

m ak minus2fprime ak( 1113857

fPrime ak( 1113857

m2 ak minusfprime ak( 1113857

fPrime ak( 1113857

m3 m2 minus2fprime m2( 1113857

fPrime m2( 1113857

(5)

JudgingWhen the distance between m2 and m3 is less than theaccuracy (|m3 minus m2|le δ) then outputxlowast (m2 + m3)2When m3 gtm2 let the abscissa of the iterative insertionpoint be as equation (2) otherwise (when m3 ltm2)exchange the values of m2 and m3 and let the iterativeinsertion pointrsquos abscissa be equal to equation (2)Calculate the values of functions f(ξk) andf(ηk) go tothe fourth step-e fourth stepWhen f(ξk)gtf(ηk) let the iterative insertion pointrsquosabscissa be equal to equation (3)Calculate the values of functions f(ξk+1) and f(ηk+1)Otherwise when f(ξk)lef(ηk) let the iterative in-sertion pointrsquos abscissa be equal to equation (4)Calculate the values of functions f(ξk+1) and f(ηk+1)Let k k + 1 go to the second step-e fifth stepWhen f(ak)gtf(bk) let the abscissas of point m m2and m3 be

m bk minus2fprime bk( 1113857

fPrime bk( 1113857

m2 ak minusfprime bk( 1113857

fPrime bk( 1113857


fPrime m2( 1113857

(6)

When the distance between m2 and m3 is less than theaccuracy (|m3 minus m2|le δ) then outputxlowast (m2 + m3)2When m3 gtm2 let the abscissa of the iterative insertionpoint be as equation (2) otherwise (when m3 ltm2)exchange the values of m2 and m3 and let the iterativeinsertion pointrsquos abscissa be equal to equation (2)Calculate the values of functions f(ξk) and f(ηk) go tothe fourth step

4 Verification Analysis

For testing the calculation effect of the new one-dimensionalsearch algorithm a calculation example f(x) x3 minus 11x2 +

4x + 60 is used to compare the calculation accuracy anditerative speed Common one-dimensional search methodsinclude Fibonacci method and Newton method Using thenew one-dimensional search method and other three kindsof methods to search the minimum points of the functionthe calculation results of various methods are shown inTable 1

Figure 2 shows that the calculated value of the GoldenSection method iteration process fluctuates around the exactvalue in the early stage As the iterative process proceeds thecalculated value finally converges to the minimum value andbasically coincides with the exact value while the iterativeprocess is generally stable However the number of itera-tions is the largest and the convergence speed is the slowestamong the new algorithm Fibonacci method and Newtonmethod Besides Golden Section methodrsquos relative errordecreases less significantly as the accuracy requirementsincrease-e Fibonacci method also has large fluctuations inthe early iteration process After nine iterations the cal-culation results converge to a minimum value and the it-erative calculation process is stable In addition the numberof iterations of the Fibonacci method is roughly equivalentto the Golden Section method but the initial convergencespeed is slower than the Golden Section method-e relativeerror did not change significantly with the increase in ac-curacy requirements but it is still better than the GoldenSection method When the position of the initial point ofNewton method is far from the minimum point in theinterval its convergence rate may be slow even not con-verge or may converge to a nonlocal minimum point When

Y

X0 ak bkm3x m2 m

m3

m2

m

f (x)

Figure 1 Schematic diagram of the new one-dimensional searchalgorithm


Table 1 Comparison of the new algorithm with the Golden Section method Fibonacci method and Newton method

Precision 01 001First try interval [1 10] [4 10]

New search algorithmNumber of iterations 3 6

Output result 7144895841400396 7146757361917681Relative error 0000262076 0000001606

Fibonacci methodNumber of iterations 11 14


Golden Section methodNumber of iterations 11 15


Newton methodNumber of iterations 3 7

Output result 0186563842345446 7146768836448334Relative error 0973895358 202637E-11

lowastTrue value is 7146768836303518 (15 decimal places are reserved)

Accuracyδ=01

True value

690

694

698

702

706

710

714

718

Out

put r

esul

t

1 2 3 40Iteration times

(a)

Accuracyδ=001

True value

65

70

75

80

85

90

95

100O

utpu

t res

ult

1 2 3 4 5 6 70Iteration times

(b)

Accuracyδ=01

True value

60

62

64

66

68

70

72

74

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(c)

Accuracyδ=001

True value

69

71

73

75

77

79

81

83

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Iteration times

(d)

Figure 2 Continued


the initial point in the interval is 1 the calculated result is farfrom the true value and the stability is poor However whenthe interval is selected properly its convergence is good andthe stability is highly accurate at 001 After four iterationsthe calculated value converges to the exact value -e iter-ation rate is increased by more than 50 compared to boththe Golden Section method and Fibonacci method -erelative error is the smallest of the four methods mainlybecause the Newton method has a local second-orderconvergence speed -e new one-dimensional search iter-ative algorithm converges to a minimum value after threeiterations Compared with the Golden Section method andthe Fibonacci method the new iterative algorithm improvedthe iteration time by 75 It also has the advantage of fastconvergence speed and stable convergence when comparedwith Newton method and it can maintain a faster con-vergence speed even under different accuracy requirementsFurthermore as accuracy requirements increase the abso-lute error decreases significantly

In general when the accuracy is set to 01 the new al-gorithm requires three iterations to obtain the target value

the Golden Section method requires 11 iterations and theFibonacci method also requires 11 iterations -e Newtonmethod cannot obtain the target value When the accuracy isset to 001 6 iterations are needed for the new algorithm toobtain the target value 15 iterations for the Golden Sectionmethod 14 iterations for the Fibonacci method and 7 it-erations for the Newton method -ere is no doubt that thenew one-dimensional search iterative algorithm has theadvantages of fast iteration speed good convergence andhigh stability of calculation results

5 Case Analysis

As shown in Figure 3 T-section beam (also called T-beam orT-shaped beam) is widely used in engineering structures forexample T-beams formed by cast-in-place rib beams andfloor slabs and independent T-beams in prefabricatedcomponents Some other prefabricated beams in the form ofcross sections such as I-beam crane beams and thin-beltroof beams are also considered according to the T-sectionbeam members Compared with the rectangular section

Accuracyδ=01

True value

65

66

67

68

69

70

71

72

73

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(e)

Accuracyδ=001

True value

70

71

72

73

74

75

76

77

78

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150Iteration times

(f )

Accuracyδ=01 True value


0

1

2

3

4

5

6

7

8

Out

put r

esul

t

(g)

Accuracyδ=001

True value6

8

10

12

14

16

18

20

22

24

Out

put r

esul

t

1 2 3 4 5 6 7 80Iteration times

(h)

Figure 2 -e relationship between the number of iterations of various algorithms and the true value (a) New algorithm iteration results atan accuracy of 01 (b) New algorithm iteration results at an accuracy of 001 (c) Golden Section method results at an accuracy of 01(d) Golden Section method results at an accuracy of 001 (e) Fibonacci method results at an accuracy of 01 (f ) Fibonacci method results atan accuracy of 001 (g) Newton method results at an accuracy of 01 (h) Newton method results at an accuracy of 001


beam the T-section beamrsquos ultimate bearing capacity is notaffected and it saves concrete materials reduces its ownweight and has certain economic benefits -is example isbased on the relevant literature and Chinese national codeGB50010-2010 ldquoSpecifications for the Design of ConcreteStructurerdquo taking the lowest cost as the goal and factoring inthe strength and structural requirements the section opti-mization design of the T-shaped reinforced concrete beam isthe preliminary design for reference

51 Establishment of Mathematical Model Assume that thespan of a simply supported beam is l the maximum bendingmoment of the interface under load is Mmax and themaximum shear force is Vmax Considering that the sectionmeets the strength conditions and the structural require-ments of the beam in the code the optimization is to makethe T-section beam with the least amount of concrete andsteel regardless of the cost of manual production

Take the price C of the beam of unit length as the ob-jective function

C ch b h0 + as( 1113857 + bfprime minus b1113872 1113873hf

prime1113960 1113961 + csAs (7)

where ch is the unit price of concrete (refer to Xirsquoan concreteprice) cs is the rebar unit price (refer to Xirsquoan rebar price) h0 isthe effective height of T-section beam hf

prime is the flange heightAs is the rebar cross-sectional area b is the web width as is theprotective layer thickness and bf

prime is the flange width

52 Constraints Flexural strength requirementsFor type I sections

Mle α1fcbfprimex h0 minus

x

21113874 1113875 (8)

where M is the design value of bending moment α1 isthe calculation coefficient when the concrete not ex-ceeding C50 10 is used and when it is C80 take thevalue as 094 and the middle-grade concrete is de-termined by linear interpolation fc is the design value

of concrete axial compressive strength and x is theheight of the interface compression zoneFor type II sections

Mle α1fc bfprime minus b1113872 1113873hf

prime h0 minushfprime

21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)

-e meaning of each coefficient in the formula is thesame as formula (8)Shear strength requirementsFor T-section or I-section simply supported beams

Vle 03βcfcbh0 (10)

where βc is concrete strength influence coefficientwhen the concrete strength grade does not exceed C50it is taken as 10 and when the concrete strength gradeis C80 it is taken as 08 and the middle-grade concreteis determined by linear interpolation

53 Reinforcement Limitation RequirementsReinforcement ratioMu shouldmeet the following equation

Mu αsbα1fcbh20 asymp 04α1fcbh

20 (11)

where αsb is the maximum resistance to bending momentcoefficient of the section -e meaning of the other coeffi-cients in the formula is the same as the above formula

Furthermore the minimum reinforcement ratio ρminshould also meet the following equation

As

bh0ge ρmin (12)

54 Simplification of the Optimization Model Since theoptimization design of reinforced concrete beams is amultivariable multiconstrained and nonlinear optimizationproblem if starting from the actual engineering based on

bfprime

hfprime

h0

as

As

b

h

Figure 3 T-shaped section in engineering structure


the analysis of design and use experience some variables willbe used as predetermined parameters to reduce the numberof design variables -e calculation of certain parameters inthe objective function and constraint conditions is simplifiedto reduce the degree of nonlinearity which makes the op-timization design problem simple and easy -erefore thisarticle deals with the optimization design of reinforcedconcrete T-section beams as follows

(1) Beam web section width b in general engineeringdesign it is usually selected according to structuralrequirements -erefore this paper considers it as apredetermined parameter and not as a designvariable

(2) In order to reduce the design variables the opti-mization is based on the rectangular cross section ofa single reinforcement -e effect of the reinforce-ment in the compression zone is not considered andonly the vertical reinforcement according to thestructure is considered

(3) In order to reduce the optimization parameters inthe constraint condition of the shear capacity of theinclined section the bending reinforcement is notconsidered only the shear effect of the stirrup (Asv)

is considered and the stirrup is not optimized as adesign variable

(4) Restrictions on crack development width crack re-sistance and deflection are not considered as con-straints As for general reinforced concrete beamsthe above conditions can basically meet the re-quirements so the impact on the optimization de-sign results is not great but it makes the optimizationprocess simpler

(5) In order to further simplify the optimization processand reduce design variables the cross-sectional areaof the cross-section steel bar when the aspect ratio ofthe web section is not greater than 25 is replaced bythe approximate formula [25]

As M

csfyh0 (13)

where cs is the internal force arm coefficient and it istaken as 09

After the above treatment the optimization designproblem of reinforced concrete T-shaped beams becomeseven simpler In the objective function this article onlyconsiders the main comparable factors that affect the cost ofreinforced concrete T-section beams that is only the cost oftensile steel and concrete in the beam is included As forother factors such as stirrup structural reinforcementconcrete formwork and labor costs after the structuraldesignplan is determined these factors have little effect onthe cost of reinforcedconcrete beams and should not becounted If it is strictly required when considering the cost

of the beam and calculating its cost it can be reflected in theunit price of concrete and steel bar in the tensile zone and itcan be converted into the relevant unit price

From formulas (11) and (12) the lower limit of theeffective height h0 is as follows

h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)

From formula (12) the lower limit of reinforcementcross-sectional area As is as follows

As ge bh0ρmin (15)

Substituting formula (12) into (6) we obtain the lowestcost function

minC ch b h0 + as( 1113857 + bfprime minus b1113872 1113873hf

prime1113960 1113961 + cs

Mmax

csfyh0 (16)

55 Instance Import -e main beam of a T-shaped beam ofa monolithic rib beam floor is known as bf

prime 2200mmhfprime 80mm and b 300mm and the environmental cate-

gory is a category as 25mm choose C30 concrete Forconcrete set the unit price of concrete at 350 yuanm3 theunit price of steel bar at 31000 yuanm3 and HRB335 gradesteel bar Mmax 400KN middot m and Vmax 450KN middot m and tryto optimize the design of the section with the lowest cost

Substituting the above known data into formula (16)gives

minC 0105h0

+ 55825 +45926

h0 (17)

MATLAB has powerful numerical calculation functionswhich can be used to solve the one-dimensional optimi-zation problem -e objective function in the problem issolved by a new one-dimensional search iterative algorithmprogramming operation and compared with the GoldenSection method

-e flowchart of the new algorithm is shown as Figure 4Table 2 shows that the calculation result of the new one-

dimensional search algorithm is h0= 66138mm (take two

significant digits) and the cost objective functionminC= 1947 yuan is taken into consideration Consideringthat the actual project can take h

0= 675mm the actual

height of the T-section beam plus the thickness of theprotective layer is 700mm and the corresponding cost is 195yuan per meter

From the calculation results in Table 2 it is not difficultto see that the Golden Section method has a small number ofiterations when the accuracy requirement is 10 and iterationtimes are 13 and stably converge to the target value even-tually when the accuracy requirement is set to 01 thecalculation result is closer to the true value However thenumber of iterations increased significantly to 18 -e newone-dimensional search method has three iterations when


the accuracy of the calculation of the objective function is10 which is only 25 iteration time of the Golden Sectionmethod which is much better than the latter -e accuracyof the calculation result is higher than that of the GoldenSection method It meets the actual requirements of theproject when the accuracy requirement is 10 the number of

iterations is 4 times and the accuracy requirements areimproved without a significant increase in the number ofiterations In general the new one-dimensional search it-erative algorithm has fast iteration speed and accuratecalculation results which greatly reduce the calculationworkload -e calculation results can be used as a reference

Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3

Swap the valueof m2 and m3

fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)

fl=f (ζk) fu=f (ηk)

a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N

m=bkmdash2f rsquo (bk)f rdquo (bk)

m3=m2mdash2f rsquo (m2)f rdquo (m2)

m2=bkmdashf rsquo (bk)f rdquo (bk)

m=akmdash2f rsquo (ak)f rdquo (ak)


m2=akmdashf rsquo (ak)f rdquo (ak)

Figure 4 Algorithm program flowchart


for the initial value of the optimal design of the T-beamsection in actual engineering

6 Conclusion

Aiming at the Golden Section method in the one-dimen-sional search method this paper proposes a new one-di-mensional search iterative method by analysing the principleof Golden Section method -e numerical example provesthat the new method has faster iteration speed than theclassic Newton method Golden Section method andFibonacci method -e number of iterations is only 25 ofthe Golden Section method It has better convergence andthe calculation is stable and reliable -e advantage is moreobvious and the error is smaller Finally an engineeringexample on the optimization design of the T-shaped rein-forced concrete beam section was verified and the followingconclusions were reached

(1) Under the same accuracy requirements the newalgorithm accelerates the interval convergence speedreduces the number of iterations and has goodnumerical stability

(2) When the accuracy of the extreme point of thefunction being calculated is not high the calculationtime can be saved and the amount of calculation canbe reduced

(3) In the engineering example of the optimal design ofthe T-shaped concrete beam the new one-dimen-sional search calculation result is more accurate thenumber of iterations is relatively small and it hascertain theoretical and practical application value

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is study was funded by the Yulin Cityrsquos 2020 Science andTechnology Plan Project (Nos CXY-2020-081 and CXY-

2020-080) and the National Natural Science Foundation ofChina Youth Fund Project (No 51808445)

References

[1] H Chen M Wang and X Zhao ldquoA multi-strategy enhancedsine cosine algorithm for global optimization and constrainedpractical engineering problemsrdquo Applied Mathematics andComputation vol 369 no 1 Article ID 124872 2020

[2] A Ahmad F Behnam S Polat A Asghar Heidari H Chenand R Tiwari ldquoMulti-strategy Gaussian Harris hawks opti-mization for fatigue life of tapered roller bearingsrdquo Engi-neering with Computers 2021

[3] J Too L Guoxi and C Huiling ldquoMemory-based Harris hawkoptimization with learning agents a feature selection ap-proachrdquo Engineering with Computers 2021

[4] B Nautiyal R Prakash V Vimal L Guoxi and C HuilingldquoImproved Salp Swarm Algorithm with mutation schemes forsolving global optimization and engineering problemsrdquo En-gineering with Computers 2021

[5] S Guo X Zhang Y Du Y Zheng and Z Cao ldquoPath planningof coastal ships based on optimized DQN reward functionrdquoJournal of Marine Science and Engineering vol 9 no 2 p 2102021

[6] W Deng J Xu H Zhao and Y Song ldquoA novel gate resourceallocation method using improved PSO-based QEArdquo IEEETransactions on Intelligent Transportation Systems vol 1-92020

[7] T Jin H Ding H Xia and J Bao ldquoReliability index and Asianbarrier option pricing formulas of the uncertain fractionalfirst-hitting time model with Caputo typerdquo Chaos Solitons ampFractals vol 142 Article ID 110409 2021

[8] T Jin H Ding L Bo X Hongxuand and X ChenxialdquoValuation of interest rate ceiling and floor based on theuncertain fractional differential equation in Caputo senserdquoJournal of Intelligent and Fuzzy Systems vol 40 no 3 2021

[9] T Jin H Xia and H Chen ldquoOptimal control problem of theuncertain second-order circuit based on first hitting criteriardquoMathematical Methods in the Applied Sciences vol 44 2021

[10] T Jin H Ding H Xia and J Bao ldquoReliability index and Asianbarrier option pricing formulas of the uncertain fractionalfirst-hitting time model with Caputo typerdquo Solitons ampFractals vol 21 no 9 2020

[11] S Qin ldquoOptimization design and information construction oflandslide engineering treatmentrdquo Chinese Journal of Geo-logical Hazard and Control vol 2-10 1999

[12] Y Liu ldquoOptimal design of reinforced concrete beam sectionrdquoFujian Architecture vol 13 pp 38ndash37 1997

[13] C Lao ldquoOptimal design of reinforced concrete beamrdquo Non-ferrous Metal Design vol 33-38 1996

Table 2 Program running results

Precision 1 01

New algorithm

First try interval [500 750] [500 750]Number of iterations 3 4

Output result 6613765 6613765Absolute error 00215 00215

Golden Section method



lowast-e true value of the number calculation is about 6613550 (4 decimal places are reserved)


[14] M Yu and Q Li ldquoOptimal design of reinforced concretesingle reinforced beamsrdquo Industrial Architecture vol 13pp 48ndash51 2000

[15] F Zhang ldquoHigh-accuracy method for calculating correlatedcolor temperature with a lookup table based on golden sectionsearchrdquo Optik vol 193 Article ID 163018 2019

[16] A Kheldoun R Bradai R Boukenoui and A Mellit ldquoA newgolden section method-based maximum power point trackingalgorithm for photovoltaic systemsrdquo Energy Conversion andManagement vol 111 pp 125ndash136 2016

[17] F Z Zoubiri R Rihani and F Bentahar ldquoGolden sectionalgorithm to optimise the chemical pretreatment of agro-industrial waste for sugars extractionrdquo Fuel vol 266 ArticleID 117028 2020

[18] G Gao N Sun S Dong Y Tao and W Wu ldquoA unifiedanalysis for the free vibration of the sandwich piezoelectriclaminated beam with general boundary conditions under thethermal environmentrdquo Shock and Vibration vol 2021 ArticleID 1328886 21 pages 2021

[19] W Shi Y Liu H Gong and C Li ldquoApplication of goldensection method in unconstrained multivariate optimizationproblemsrdquo Journal of Northeast Normal University vol 35no 3 pp 11ndash14 2003

[20] X Liu Z Mei and S Zhang ldquo618 method Fibonacci methodand parabola method to search for extreme points of thecompound pendulum periodrdquo Physics Experimentationvol 30 pp 35ndash38 2010

[21] R Qian ldquoA method for optimizing smoothing coefficientusing 0618 optimization methodrdquo Systems Engineeringvol 59-61 1994

[22] J Zhang and J Chen ldquoTwo fast one-dimensional searchmethodsrdquo in Proceedings of the Seventh Academic ExchangeConference of China Operations Research Society vol 1pp 73ndash78 Wuhan China 2004

[23] Z Zhang F Jin and Y Tang ldquoAn improved one-dimensionalsearch index optimization algorithmrdquo Journal of HuaqiaoUniversity vol 33 pp 503ndash505 2012

[24] Y Liu ldquoSome discussions on the golden section methodrdquoElectromechanical Technologyrdquo vol 13-14+59 2006

[25] Y Li X Qin H Cong and M Luo ldquoResearch on optimumdesign of sections of large aspect ratio concrete independentfoundation sectionrdquo Industrial Architecture vol 48pp 128ndash132+168 2018


addition Jin et al used mathematical models to optimize thedesign of financial market models and circuit control sys-tems [7ndash10] Seeking the lowest cost design under thepremise of meeting and application is to optimize the design-e design principle is the optimal design the designmethod is the computer and the design method is theoptimal mathematical method [11] Structural optimizationhas been adopted in many structural engineering practicessuch as quilted structures frame structures bridges hy-draulic structures and other optimization designs that havereceived good results Many scholars have conducted a lot ofresearch on the structural design of reinforced concretebeams Liu [12] used the method of structural optimizationto determine the section height of reinforced concrete beamsand changed the traditional iterative calculation method ofdetermining the beam heightso that the beam height wasoptimized to simplify the design and reduce the project costLao [13] considered the load-bearing capacity requirementsand reinforcement structure of the reinforced concrete beamsection to optimize the design of the beam section anddiscussed the application of the 0618 method in the opti-mization design Yu and Li [14] used the improved con-strained nonlinear mixed discrete variable optimizationdesign method (MDOD) to optimize the design of rein-forced concrete single-reinforced beams taking the cost ofbeam unit length as the objective function and the strengthof positive and oblique sections and the structural re-quirements of single-reinforced beams as the constraintconditions

One-dimensional search also known as linear searchrefers to the optimization of a single-variable function and isone of the basic methods for solving unconstrained non-linear programming problems One-dimensional search canbe used independently to solve single-variable optimizationproblems and at the same time it is a common method tosolve multivariable optimization problems Although one-dimensional search itself can be used to solve many one-dimensional optimization problems in science and practiceit is more used as a means of accelerating algorithm con-vergence and is used in conjunction with multidimensionaloptimization methods such as those suitable for solvinglarge-scale unconstrained optimization -e conjugategradient method of the problem is combined with variousquasi-Newton methods to solve numerous multidimen-sional optimization problems As one of the one-dimen-sional search methods the Golden Section method has highaccuracy and does not require high requirements for thedifferentiability and convergence of the function -eGolden Section method has a wide range of applicabilityZhang [15] built a high-precision correlated colour tem-perature calculation method based on a small-scale lookuptable based on the Golden Section method and the resultsshowed that the algorithm has excellent performanceKheldoun et al [16] proposed a new method for tracking themaximum power point of photovoltaic systems based on theGolden Section method optimization technology whichensures that the photovoltaic system can quickly converge tothe maximum power point thereby reducing energy wasteZoubiri et al [17] used the Golden Section method to

optimize the chemical pretreatment of agroindustrial wasteused to extract sugar and the results showed that the GoldenSection algorithm is a useful step to ensure the extraction oftotal sugar Gao et al [18] studied the free vibration char-acteristics of sandwich piezoelectric beams under elasticboundary conditions and thermal environments and in-troduced a search algorithm based on the Golden Sectionmethod search to calculate the required frequency param-eters Finally the correctness and universality of the algo-rithm were verified by simulation software Shi et al [19]applied the Golden Section one-dimensional search methodto unconstrained multivariate optimization problem solv-ing compared it with Newtonrsquos method and dampedNewtonrsquos method and proved that the Golden Section al-gorithm is effective and practical Liu et al [20] used theldquo0618 methodrdquo the Fibonacci method and the parabolamethod to search for the extreme points of the pendulumcompound pendulum period -rough experiments it isfound that the ldquo0618 methodrdquo used 15 times to measure andthe Fibonacci method used 17 times to measure -eFibonacci method has higher search accuracy than theldquoGolden Section methodrdquo but it does not have an advantageof search speed in comparison However some scholarsbelieve that the Golden Section method had shortcomingsQian [21] pointed out that using the Golden Section methodto optimize can only be a unimodal function in the intervaland the scope of application is more limited Zhang andChen [22] believe that the Golden Section method is rep-resented by the division method Although this type ofmethod has global convergence it does not use the prop-erties of the function For some functions with better an-alytical properties the convergence speed of this method istoo slow Zhang et al [23] proposed an improved one-di-mensional search index optimization algorithm -e ex-ample verification results show that the index optimizationmethod is faster than the Golden Section method At thesame time the interval accuracy of the optimal solution ismore accurate and when the solution intervals are similarthe convergence speed is still faster than that of the GoldenSection method Since the Golden Section method shortensthe length of the search interval by 618 each time Liu et al[24] believe that from the programming perspective theGolden Section method only needs to insert one point at atime but the interval shortening rate is fixed at 618-erefore a midpoint method was proposed to increase theshortening rate to 51 -is paper proposes a new one-dimensional search algorithm based on the iterative algo-rithm of the Golden Section method (Algorithm 1)

2 Background Information

-e Golden Section method is one of the main methods foraccurate one-dimensional search By taking test points andcomparing function values the search interval containingthe minimum points is continuously shortened When theinterval length is shortened to a certain extent the points onthe interval all are close to the minimum so each point onthe interval can be regarded as an approximation of theminimum point




λk ak + 0382 bk minus ak( 1113857

μk ak + 0618 bk minus ak( 1113857(1)






and ηk and let

ξk m2 + 0382 m3 minus m2( 1113857

ηk m2 + 0618 m3 minus m2( 1113857(2)


ak+1 ξk

bk+1 m3

ξk+1 ηk

ηk+1 ak+1 + 0618 bk+1 minus ak+1( 1113857

(3)



ak+1 m2

bk+1 ηk

ηk+1 ξk

ξk+1 ak+1 + 0382 bk+1 minus ak+1( 1113857

(4)














fPrime ak( 1113857


fPrime ak( 1113857


fPrime m2( 1113857

(5)



fPrime bk( 1113857


fPrime bk( 1113857


fPrime m2( 1113857

(6)






Y

X0 ak bkm3x m2 m

m3

m2

m

f (x)














Accuracyδ=01

True value

690

694

698

702

706

710

714

718

Out

put r

esul

t


(a)

Accuracyδ=001

True value

65

70

75

80

85

90

95

100O

utpu

t res

ult


(b)

Accuracyδ=01

True value

60

62

64

66

68

70

72

74

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(c)

Accuracyδ=001

True value

69

71

73

75

77

79

81

83

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Iteration times

(d)

Figure 2 Continued





5 Case Analysis


Accuracyδ=01

True value

65

66

67

68

69

70

71

72

73

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(e)

Accuracyδ=001

True value

70

71

72

73

74

75

76

77

78

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150Iteration times

(f )



0

1

2

3

4

5

6

7

8

Out

put r

esul

t

(g)

Accuracyδ=001

True value6

8

10

12

14

16

18

20

22

24

Out

put r

esul

t


(h)







prime1113960 1113961 + csAs (7)






x

21113874 1113875 (8)





21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)


Vle 03βcfcbh0 (10)




20 (11)



As

bh0ge ρmin (12)


bfprime

hfprime

h0

as

As

b

h










As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm























λk ak + 0382 bk minus ak( 1113857

μk ak + 0618 bk minus ak( 1113857(1)






and ηk and let

ξk m2 + 0382 m3 minus m2( 1113857

ηk m2 + 0618 m3 minus m2( 1113857(2)


ak+1 ξk

bk+1 m3

ξk+1 ηk

ηk+1 ak+1 + 0618 bk+1 minus ak+1( 1113857

(3)



ak+1 m2

bk+1 ηk

ηk+1 ξk

ξk+1 ak+1 + 0382 bk+1 minus ak+1( 1113857

(4)














fPrime ak( 1113857


fPrime ak( 1113857


fPrime m2( 1113857

(5)



fPrime bk( 1113857


fPrime bk( 1113857


fPrime m2( 1113857

(6)






Y

X0 ak bkm3x m2 m

m3

m2

m

f (x)














Accuracyδ=01

True value

690

694

698

702

706

710

714

718

Out

put r

esul

t


(a)

Accuracyδ=001

True value

65

70

75

80

85

90

95

100O

utpu

t res

ult


(b)

Accuracyδ=01

True value

60

62

64

66

68

70

72

74

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(c)

Accuracyδ=001

True value

69

71

73

75

77

79

81

83

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Iteration times

(d)

Figure 2 Continued





5 Case Analysis


Accuracyδ=01

True value

65

66

67

68

69

70

71

72

73

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(e)

Accuracyδ=001

True value

70

71

72

73

74

75

76

77

78

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150Iteration times

(f )



0

1

2

3

4

5

6

7

8

Out

put r

esul

t

(g)

Accuracyδ=001

True value6

8

10

12

14

16

18

20

22

24

Out

put r

esul

t


(h)







prime1113960 1113961 + csAs (7)






x

21113874 1113875 (8)





21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)


Vle 03βcfcbh0 (10)




20 (11)



As

bh0ge ρmin (12)


bfprime

hfprime

h0

as

As

b

h










As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm






















fPrime ak( 1113857


fPrime ak( 1113857


fPrime m2( 1113857

(5)



fPrime bk( 1113857


fPrime bk( 1113857


fPrime m2( 1113857

(6)






Y

X0 ak bkm3x m2 m

m3

m2

m

f (x)














Accuracyδ=01

True value

690

694

698

702

706

710

714

718

Out

put r

esul

t


(a)

Accuracyδ=001

True value

65

70

75

80

85

90

95

100O

utpu

t res

ult


(b)

Accuracyδ=01

True value

60

62

64

66

68

70

72

74

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(c)

Accuracyδ=001

True value

69

71

73

75

77

79

81

83

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Iteration times

(d)

Figure 2 Continued





5 Case Analysis


Accuracyδ=01

True value

65

66

67

68

69

70

71

72

73

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(e)

Accuracyδ=001

True value

70

71

72

73

74

75

76

77

78

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150Iteration times

(f )



0

1

2

3

4

5

6

7

8

Out

put r

esul

t

(g)

Accuracyδ=001

True value6

8

10

12

14

16

18

20

22

24

Out

put r

esul

t


(h)







prime1113960 1113961 + csAs (7)






x

21113874 1113875 (8)





21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)


Vle 03βcfcbh0 (10)




20 (11)



As

bh0ge ρmin (12)


bfprime

hfprime

h0

as

As

b

h










As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm
































Accuracyδ=01

True value

690

694

698

702

706

710

714

718

Out

put r

esul

t


(a)

Accuracyδ=001

True value

65

70

75

80

85

90

95

100O

utpu

t res

ult


(b)

Accuracyδ=01

True value

60

62

64

66

68

70

72

74

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(c)

Accuracyδ=001

True value

69

71

73

75

77

79

81

83

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Iteration times

(d)

Figure 2 Continued





5 Case Analysis


Accuracyδ=01

True value

65

66

67

68

69

70

71

72

73

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(e)

Accuracyδ=001

True value

70

71

72

73

74

75

76

77

78

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150Iteration times

(f )



0

1

2

3

4

5

6

7

8

Out

put r

esul

t

(g)

Accuracyδ=001

True value6

8

10

12

14

16

18

20

22

24

Out

put r

esul

t


(h)







prime1113960 1113961 + csAs (7)






x

21113874 1113875 (8)





21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)


Vle 03βcfcbh0 (10)




20 (11)



As

bh0ge ρmin (12)


bfprime

hfprime

h0

as

As

b

h










As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm
























5 Case Analysis


Accuracyδ=01

True value

65

66

67

68

69

70

71

72

73

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 120Iteration times

(e)

Accuracyδ=001

True value

70

71

72

73

74

75

76

77

78

Out

put r

esul

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150Iteration times

(f )



0

1

2

3

4

5

6

7

8

Out

put r

esul

t

(g)

Accuracyδ=001

True value6

8

10

12

14

16

18

20

22

24

Out

put r

esul

t


(h)







prime1113960 1113961 + csAs (7)






x

21113874 1113875 (8)





21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)


Vle 03βcfcbh0 (10)




20 (11)



As

bh0ge ρmin (12)


bfprime

hfprime

h0

as

As

b

h










As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm

























prime1113960 1113961 + csAs (7)






x

21113874 1113875 (8)





21113888 1113889 + α1fcbx h0 minus

x

21113874 1113875

(9)


Vle 03βcfcbh0 (10)




20 (11)



As

bh0ge ρmin (12)


bfprime

hfprime

h0

as

As

b

h










As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm




























As M

csfyh0 (13)





h0 maxVmax

03βcfcbh0

Mmax

04α1fcb

1113971

⎛⎝ ⎞⎠ (14)


As ge bh0ρmin (15)



prime1113960 1113961 + cs

Mmax

csfyh0 (16)





minC 0105h0

+ 55825 +45926

h0 (17)





0= 675mm the actual






Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm























Input h0h1 δ

fa=f (h0)=f (h1)

falt

m2ltm3


fultfl

ζk= m2 + 0382 (m3-m2)ηk= m2 + 0618 (m3 - m2)


a=ζk b=m3ζk+1=ηk

ηk+1=a+0618 (b-a)

a=m2 b=ηkηk+1=ζk

ζk+1=a+0382 (b-a)

|a-b| le δ

optimal value

YN

Y

N

Y N

Y

N










6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm






















6 Conclusion





Data Availability




Acknowledgments



References















Precision 1 01

New algorithm


































Documents

New One-Dimensional Search Iteration Algorithm and