a report on numeric benchmark functions · 2020. 9. 10. · a report on numeric benchmark functions...

Center of Excellence on Soft Computing and Intelligent Information Processing

A Report on

Numeric Benchmark Functions

Version 1.0: 5/20/2015

Presented by: Kayvan Nalaie

Toktam Saghafi

Faculty Advisor: Dr.M. Akbarzadeh.T

NUMERIC BENCHMARK FUNCTIONS

CENTER OF EXCELLENCE ON SOFT COMPUTING AND INTELLIGENT INFORMATION PROCESSING

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Research Objective and Methodology

This paper provides the review of literature benchmarks (test functions) commonly used in order to test optimization procedures dedicated for multidimensional,continuous optimization task. Special attention has been paid to multiple-extreme functions, treated as the quality test for “resistant” optimization methods (GA, SA, TS, etc.).

Quality of optimization procedures (those already known and these newly proposed) are frequently evaluated by using common standard literature benchmarks.



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Outline

RESEARCH OBJECTIVE AND METHODOLOGY 2

OUTLINE 3

NUMERIC B.FS 8

Convex and Multimodal Functions 9

MULTIVARIATE B.FS 10

Bowl-Shaped B.F 10 De Jong’s B.F 10 Axis parallel hyper-ellipsoid B.F 11 Rotated hyper-ellipsoid B.F 12 Sum of different power B.F 13 Schwefel 1 B.F 14 Schwefel 21 B.F 15 Damavandi B.F. 16 Exponential B.F 17 Wayburn and Seader 2 B.F 18

Multimodal B.F 20 Rastrigin’s B.F 20 Griewangk B.F 21 Langermann B.F 22 Shubert4 B.F 23 Griewank B.F 24 Levy13 B.F 25 Alpine 1 B.F 26 Alpine 2 B.F 27 Giunta B.F 28 Keane B.F 29 Jennrich-Sampson B.F 30 Hosaki B.F 31 Bird B.F 32 Rastrigin B.F 33 CarromTable B.F 34 Chichinadze B.F 35 Cosine Mixture B.F 36 Egg Crate B.F. 37 Levy 5 B.F 38 Schaffer 1 B.F 39 Schaffer 3 B.F 40 Salomon B.F. 41 Whitley B.F 42 Weierstrass B.F 43



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W / Wavy B.F 44 Levy13 B.F 45 Egg Holder B.F 46 Cross-in-Tray B.F 47 Crowned Cross B.F 48 Deb 1 B.F 49 Deb 2 B.F 50 DeflectedCorrugatedSpring B.F 51 Mishra 3 test objective function. 52 Xin-She Yang 4 test objective function. 53

Valley-Shaped B.F 54 Wayburn and Seader 1 B.F 54 Rosenbrock’s valley B.F 55 Michalewicz B.F 56 Six-hump camel back B.F 57 Deceptive B.F 58 AMGM B.F 60 Leon B.F 61 Goldstein-Price B.F 62 Judge B.F 63 Bartels-Conn B.F 64 El-Attar-Vidyasagar-Dutta B.F 65 Beale B.F 66 Bohachevsky B.F 67 Brown Test Objective Function 68 Bukin 2 B.F 69 Bukin 4 B.F 70 Deckkers-Aarts B.F 71 Bukin 6 B.F 72 Cigar B.F. 73 Csendes B.F 74 HimmelBlau B.F 75 Decanomial B.F 76 Deckkers-Aarts B.F 77 Dixon and Price B.F 78 McCormick test objective function. 79 Six Hump Camel test objective function. 80

Steep Ridges/Drops-Shaped B.F 81 Ackley’s B.F 81 Easom B.F 82 DropWave B.F 83 Shekel105 B.F 84 Cross-Leg-Table B.F 85 Xin-She Yang 3 test objective function. 86

Plate-Shaped B.F 87 Branins B.F 87 Goldstein-Price B.F 88 Adjiman B.F 89 Brent B.F 90 Cube B.F 91 Katsuura B.F 92



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Exp2 B.F 93

UNIVARIATE TEST FUNCTIONS 94 Univariate Problem02 B.F 94 Univariate Problem03 B.F 95 Univariate Problem04 B.F 96 Univariate Problem05 B.F 97 Univariate Problem06 B.F 98 Univariate Problem07 B.F 99 Univariate Problem08 B.F 100 Univariate Problem09 B.F 101 Univariate Problem10 B.F 102 Univariate Problem11 B.F 103 Univariate Problem12 B.F 104 Univariate Problem13 B.F 105 Univariate Problem14 B.F 106 Univariate Problem15 B.F 107 Univariate Problem18 B.F 108 Univariate Problem20 B.F 109 Univariate Problem21 B.F 110

Univariate Problem22 B.F 111

SOURCE CODE 112

MVF Source Code 112 double mvfAckley(int n, double *x) 113 double mvfBeale(int n, double *x) 114 double mvfBohachevsky1(int n, double *x) 114 double mvfBohachevsky2(int n, double *x) 114 double mvfBooth(int n, double *x) 114 double mvfBoxBetts(int n, double * x) 115 double mvfBranin(int n, double *x) 115 double mvfBranin2(int n, double *x) 116 double mvfCamel3(int n, double *x) 116

double mvfCamel6(int n, double *x) 116 double mvfChichinadze(int n, double *x) 116 double mvfCola( int n, double * x ) 117 double mvfColville(int n, double *x) 117 double mvfCorana(int n, double *x) 118 double mvfEasom(int n, double *x) 118 double mvfEggholder(int n, double *x) 119 double mvfExp2(int n, double *x) 119 double mvfFraudensteinRoth(int n, double *x) 120 double mvfGeneralizedRosenbrock(int n, double * x) 120 double mvfGoldsteinPrice(int n, double * x) 120 double mvfGriewank(int n, double * x) 120 double mvfHansen(int n, double * x) 121 double mvfHartman3(int n, double *x) 121 double mvfHartman6(int n, double *x) 122 double mvfHimmelblau(int n, double *x) 123 double mvfHolzman1(int n, double *x) 123 double mvfHosaki(int n, double *x) 124 double mvfHyperellipsoid(int n, double *x) 124



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double mvfKatsuuras(int n, double *x) 125 double mvfKowalik(int n, double *x) 125 double mvfLangerman(int n, double *x) 127 double mvfLennardJones(int n, double *x) 127 double mvfLeon(int n, double *x) 128 double mvfLevy(int n, double * x) 128 double mvfMatyas(int n, double *x) 128 double mvfMaxmod(int n, double *x) 128 double mvfMcCormick(int n, double * x) 129 double mvfMichalewitz(int n, double *x) 129 double mvfMultimod(int n, double *x) 129 double mvfNeumaierPerm(int n, double x[]) 130 double mvfNeumaierPerm0(int n, double x[]) 130 double mvfNeumaierPowersum(int n, double x[]) 131 double mvfNeumaierTrid(int n, double x[]) 131 double mvfOddsquare(int n, double *x) 132 double mvfPaviani(int n, double *x) 132 double mvfPlateau(int n, double *x) 133 double mvfPowell(int n, double *x) 133 double mvfQuarticNoiseU(int n, double *x) 134 double mvfQuarticNoiseZ(int n, double *x) 134 double mvfRana(int n, double *x) 134 double mvfRastrigin(int n, double *x) 135 double mvfRastrigin2(int n, double *x) 135 double mvfRosenbrock(int n, double * x) 136 double mvfSchaffer2(int n, double *x) 136 double mvfSchwefel1_2(int n, double *x) 136 double mvfSchwefel2_21(int n, double *x) 137 double mvfSchwefel2_22(int n, double *x) 138 double mvfSchwefel2_26(int n, double *x) 138 double mvfShekel2(int n, double *x) 138 double mvfShekelSub4(int m, double *x) 140 double mvfShekel4_5(int n, double * x) 140 double mvfShekel4_7(int n, double * x) 140 double mvfShekel4_10(int n, double * x) 140 double mvfShekel10(int n, double *x) 141 double mvfShubert(int n, double * x) 141 double mvfShubert2(int n, double *x) 142 double mvfShubert3(int n, double *x) 142 double mvfSphere(int n, double *x) 143 double mvfSphere2(int n, double *x) 143 double mvfStep(int n, double *x) 143 double mvfStretchedV(int n, double *x) 144 double mvfSumSquares(int n, double *x) 144 double mvfTrecanni(int n, double *x) 145 double mvfTrefethen4(int n, double *x) 145 double mvfXor(int n, double* x) 146 double mvfWatson(int n, double *x) 146 double mvfZettl(int n, double *x) 146 double mvfZimmerman(int n, double *x) 147

Matlab & R Implemented Source Code 149 SPHERE FUNCTION 149 ACKLEY FUNCTION 150



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BUKIN FUNCTION N. 6 151 A. CROSS-IN-TRAY FUNCTION 152 DROP-WAVE FUNCTION 154 EGGHOLDER FUNCTION 155 GRAMACY & LEE (2012) FUNCTION 155 GRIEWANK FUNCTION 156 HOLDER TABLE FUNCTION 157 LANGERMANN FUNCTION 158 LEVY FUNCTION 161 RASTRIGIN FUNCTION 162 SCHAFFER FUNCTION N. 2 163 SCHWEFEL FUNCTION 163 SHUBERT FUNCTION 164 BOHACHEVSKY FUNCTION 3 166 PERM FUNCTION 0, d, beta 167 ROTATED HYPER-ELLIPSOID FUNCTION 168 SUM OF DIFFERENT POWERS FUNCTION 170 TRID FUNCTION 171 BOOTH FUNCTION 172

MATYAS FUNCTION 173 MCCORMICK FUNCTION 174 POWER SUM FUNCTION 175 ZAKHAROV FUNCTION 176 THREE-HUMP CAMEL FUNCTION 177 SIX-HUMP CAMEL FUNCTION 179 DIXON-PRICE FUNCTION 180 ROSENBROCK FUNCTION 181 DE JONG FUNCTION N. 5 182 EASOM FUNCTION 183 MICHALEWICZ FUNCTION 184 BEALE FUNCTION 185 BRANIN FUNCTION, MODIFIED 186 COLVILLE FUNCTION 188 GOLDSTEIN-PRICE FUNCTION 189 PERM FUNCTION d, beta 191 POWELL FUNCTION 193 STYBLINSKI-TANG FUNCTION 194

REFERENCES 196



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Numeric B.Fs

There are several classes of such test functions, all of them are continuous.

(a) Unimodal, convex, multidimensional

(b) Multimodal, two-dimensional with a small number of local extremes

(c) Multimodal, two-dimensional with huge number of local extremes

(d) Multimodal, multidimensional, with huge number of local extremes.

Class (a) contains nice functions as well as malicious cases causing poor or slow convergence to single global extremum. Class (b) is mediate between (a) and (c)- (d), and is used to test quality of standard optimization procedures in the hostile environment, namely that having few local extremes with single global one. Classes (c)-(d) are recommended to test quality of intelligent “resistant” optimization methods, as an example GA, SA, TS, etc. These classes are considered as very hard test problems. Class (c) is “artificial” in some sense, since the behavior of optimization procedure is usually being justified, explain and supported by human intuitions on 2D surface. Moreover, two-dimensional optimization problems appear very rarely in practice. Unfortunately, practical discrete optimization problems provide instances with large number of dimensions, laying undoubtedly in class (d). For example, the smallest known currently benchmark ft10 for so called job shop scheduling problem has dimension 90, the biggest known - has dimension 1980. Therefore, in order to test real quality of proposed algorithms, we need to consider chiefly instances from class (d). As the shocking contrast, the proposed GA approaches for continuous optimization do not exceed dimension 10.



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Convex and Multimodal Functions Geometrically, a function is convex if a line segment drawn from any point (x, f(x))

to another point (y, f(y)) -- called the chord from x to y -- lies on or above the graph of f, as in the picture below:

Algebraically, f is convex if, for any x and y, and any t between 0 and 1, f( tx + (1-t)y )



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Multivariate B.Fs

Bowl-Shaped B.F

De Jong’s B.F

So called first function of De Jong’s is one of the simplest test benchmark. The simplest test function is De Jong's function 1. It is also known as sphere model. It is continuos, convex and unimodal. Function is continuous, convex and unimodal. It has the following general definition Test area is usually restricted to hyphercube -5.12



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Axis parallel hyper-ellipsoid B.F

The axis parallel hyper-ellipsoid is similar to function of De Jong. It is also known as the weighted sphere model. Function is continuous, convex and unimodal. It has the following general definition

Test area is usually restricted to hyphercube -5.12



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Rotated hyper-ellipsoid B.F

An extension of the axis parallel hyper-ellipsoid is Schwefel’s function. With respect to the

coordinate axes, this function produces rotated hyper-ellipsoids. It is continuous, convex and

unimodal. Function has the following general definition

Test area is usually restricted to hyphercube −65.539 ≤ 𝑥𝑖 ≤ 65.536, i = 1,…, n. Its global

minimum equal f(x) = 0 is obtainable for xi = 0, i = 1,..,n.

Function Properties

Variation Multivariate

Shape Bowl-Shaped

Global minimum 0

Figure 5 Rotated hyper-ellipsoid function in 2D,f(x,y) = x^2 + (x^2 + y^2)



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Sum of different power B.F

The sum of different powers is a commonly used unimodal test function. It has the following definition

Test area is usually restricted to hyphercube -1 ≤xi≤ 1, i = 1,…, n. Its global minimum equal f(x)=0 is obtainable for xi = 0, i = 1,..,n.

Figure 6 Sum of different power functions in 2D,f(x,y) =|x|^2 +|y|^3

Function Properties


Shape Bowl-Shaped

Global minimum 0



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Figure 7 Two-dimensional Schwefel 1 function

Schwefel 1 B.F

This class defines the Schwefel 1 global optimization problem. This is a unimodal

minimization problem defined as follows:

Where, in this exercise, .

Here, represents the number of dimensions and for .

Global optimum: f(x_i) = 0 for x_i = 0 for i=1,...,n.

Function Properties


Shape Bowl-Shaped

Global minimum 0



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Figure 8 Two-dimensional Schwefel 21 function

Schwefel 21 B.F

This class defines the Schwefel 21 global optimization problem. This is a unimodal




Function Properties


Shape Bowl/Plate-Shaped

Global minimum 0



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Damavandi B.F.

This class defines the Damavandi global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [0, 14] for i=1,...,n.

Global optimum: f(x_i) = 0.0 for x_i = 2 for i=1,...,n.

Figure 9Two-dimensional Damavandi function

Function Properties

Variation Bivariate

Shape Bowl/Drop -Shaped

Global minimum 0



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Exponential B.F

This class defines the Exponential global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [-1, 1] for i=1,...,n.

Global optimum: f(x_i) = -1 for x_i = 0 for i=1,...,n.

Figure 10 Two-dimensional Exponential function

Function Properties

Variation Bivariate

Shape Bowl-Shaped

Global minimum -1



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Wayburn and Seader 2 B.F

This class defines the Wayburn and Seader 2 global optimization problem. This is a

unimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for x_i = [0.2, 1].

Figure 11 Two-dimensional Wayburn and Seader 2 function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



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Step test objective function

This class defines the Step global optimization problem. This is a multimodal minimization

problem defined as follows:

Here, n represents the number of dimensions and x_i ∈[-100, 100] for i=1,...,n.

Global optimum: f(x_i) = 0 for x_i = 0.5 for i=1,...,n

Figure 12 Two-dimensional Step function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



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Multimodal B.F

Rastrigin’s B.F

Rastrigin’s function is based on the function of De Jong with the addition of cosine modulation in order to produce frequent local minima. Thus, the test function is highly multimodal. However, the location of the minima are regularly distributed. Function has the following definition

Test area is usually restricted to hyphercube -5.12≤xi≤5.12, i = 1,…, n. Its global minimum equal f(x)=0 is obtainable for xi = 0, i = 1,..,n.

Figure 13 An overview of Rastrigin’s function in 2D

,f(x,y)= 10x2 + [x^2-10 cos(2πx)] + [y^2-10 cos(2πy)]

Function Properties


Shape Many Local Minima

Global minimum 0



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Griewangk B.F

Griewangk’s function is similar to the function of Rastrigin. It has many widespread local minima regularly distributed. Function has the following definition

Test area is usually restricted to hyphercube -600 ≤xi≤ 600, i = 1,…, n. Its global minimum equal f(x)=0 is obtainable for xi = 0, i = 1,..,n. The function interpretation changes with the scale, the general overview suggests convex function, medium-scale view suggests existence of local extremum, and finally zoom on the details indicates complex structure of numerous local extremum.

Figure 14 Medium-scale view of Griewangk’s function in 2D,

f(x,y)=(x^2+y^2)/4000-cos(x)cos(y/√2)+1

Function Properties



Global minimum 0



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Langermann B.F

This class defines the Langermann global optimization problem. This is a multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ∈ [0, 10] for i=1,2.

Global optimum: f(x_i) = -5.1621259 for x= [2.00299219, 1.006096]

Figure 15 Two-dimensional Langermann function

Function Properties

Variation Bivariate


Global minimum -5.16212



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Shubert4 B.F

This class defines the Shubert 4 global optimization problem. This is a multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and xi ∈ [-10, 10] for i=1,...,n.

Global optimum: f(x_i) = -29.016015 for x = [-0.80032121, -7.08350592] (and many others).

Figure 16 An overview of Shuber4 function

Function Properties






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Griewank B.F

This class defines the Griewank global optimization problem. This is a continues and multimodal minimization problem defined as follows:



Figure 17 Two-dimensional Griewank function

Function Properties

Variation Bivariate


Global minimum 0



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Levy13 B.F

This class defines the Levy13 global optimization problem. This is a continues and

multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for x_i = 1 for i=1,2

Figure 18 Two-dimensional Levy13 function

Function Properties

Variation Bivariate


Global minimum 0



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Alpine 1 B.F

his class defines the Alpine 1 global optimization problem. This is a continues and multimodal and continues and minimization problem defined as follows:

Here, n represents the number of dimensions and x_i 𝜖 [-10, 10] for i=1,...,n.


Figure 19 Two-dimensional Alpine 1 function

Function Properties



Global minimum 0



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Alpine 2 B.F

This class defines the Alpine 2 global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -6.1295 for x_i = 7.917 for i=1,...,n.

Figure 20 Two-dimensional Alpine 2 function

Function Properties


Shape Multi Local Minima




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Giunta B.F

This class defines the Giunta global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [-1, 1] for i=1,2.

Global optimum: f(x_i) = 0.06447042053690566 for X = [0.4673200277395354, 0.4673200169591304].

Figure 21 Two-dimensional Giunta function

Function Properties

Variation Bivariate


Global minimum 0.06447



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Keane B.F

This class defines the Keane global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [0, 10] for i=1,2.


Figure 22 Two-dimensional Keane function

Function Properties

Variation Bivariate





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Jennrich-Sampson B.F

This class defines the Jennrich-Sampson global optimization problem. This is a continues and multimodal minimization problem defined as follows:



Figure 233 Two-dimensional Jennrich-Sampson function

Function Properties

Variation Bivariate





PAGE 31

Hosaki B.F

This class defines the Hosaki global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -2.3458 for X = [4, 2].

Figure 24 Two-dimensional Hosaki function

Function Properties

Variation Bivariate

Shape Tow Local Minima




PAGE 32

Bird B.F

This class defines the Bird global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [-2\pi, 2\pi] for i=1,2.

Global optimum: f(x_i) = -106.7645367198034 for x = [4.701055751981055 , 3.152946019601391] or x= [-1.582142172055011, -3.130246799635430]

Figure 25 Two-dimensional Bird function

Function Properties

Variation Bivariate





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Rastrigin B.F

This class defines the Rastrigin global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ∈ [-5.12, 5.12] for i=1,...,n.


Figure 26 Two-dimensional Rastrigin function

Function Properties



Global minimum 0



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CarromTable B.F

This class defines the CarromTable global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -24.15681551650653 for x_i = +/- 9.646157266348881 for i=1,...,n.

Figure 27 Two-dimensional CarromTable function

Function Properties

Variation Bivariate





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Chichinadze B.F

This class defines the Chichinadze global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -42.94438701899098 for X = [6.189866586965680, 0.5].

Figure 28 Two-dimensional Chichinadze function

Function Properties

Variation Bivariate





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Cosine Mixture B.F

This class defines the Cosine Mixture global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [-1, 1] for i=1,...,N.

Global optimum: f(x_i) = -0.1N for x_i = 0 for i=1,...,N

Figure 29 Two-dimensional Cosine Mixture function

Function Properties



Global minimum -0.1n



PAGE 37

Egg Crate B.F.

This class defines the Egg Crate global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for x_i = 0 for i=1,2.

Figure 30 Two-dimensional Egg Crate function

Function Properties

Variation Bivariate


Global minimum 0



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Levy 5 B.F

This class defines the Levy 5 global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -176.1375 for X = [-1.3068, -1.4248].

Figure 31 Two-dimensional Levy 5 function

Function Properties

Variation Bivariate





PAGE 39

Figure 32 Two-dimensional Schaffer 1 function

Schaffer 1 B.F

This class defines the Schaffer 1 global optimization problem. This is a multimodal



Global optimum: f(x_i) = 0 for x_i = 0 for i=1,2 .

Function Properties

Variation Bivariate


Global minimum 0



PAGE 40

Schaffer 3 B.F

This class defines the Schaffer 3 global optimization problem. This is a multimodal



Global optimum: f(x_i) = 0.00156685 for x_i = [0, 1.253115].

Figure 33 Two-dimensional Schaffer 3 function

Function Properties

Variation Bivariate





PAGE 41

Salomon B.F.

This class defines the Salomon global optimization problem. This is a multimodal



Global optimum: f(x_i) = 0 for x_i = 0 for i=1,...,

Figure 34 Two-dimensional Salomon function

Function Properties



Global minimum 0



PAGE 42

Figure 35 Two-dimensional Whitley function

Whitley B.F

This class defines the Whitley global optimization problem. This is a multimodal


Here, represents the number of dimensions

and for .

Global optimum: f(x_i) = 0 for x_i = 1 for i=1,...,n .

Function Properties



Global minimum 0



PAGE 43

Figure 36 Two-dimensional Weierstrass function

Weierstrass B.F

This class defines the Weierstrass global optimization problem. This is a multimodal minimization problem defined as follows:

Where, in this exercise, kmax = 20, a = 0.5 and b = 3.

Here, n represents the number of dimensions and x_i ϵ [-0.5, 0.5] for i=1,...,n.


Function Properties



Global minimum 4



PAGE 44

Figure 37 Two-dimensional W / Wavy function

W / Wavy B.F

This class defines the W / Wavy global optimization problem. This is a multimodal


Where, in this exercise, . The number of local minima is and for

odd and even respectively.


Global optimum: f(x_i) = 0 for x_i = 0 for i=1,2 .

Function Properties

Variation Bivariate


Global minimum 0



PAGE 45

Levy13 B.F

This class defines the Levy13 global optimization problem. This is a continues and



Global optimum: f(x_i) = 0 for x_i = 1 for i=1,2

Figure 38 Two-dimensional Levy13 function

Function Properties

Variation Bivariate


Global minimum 0



PAGE 46

Egg Holder B.F

This class defines the Egg Holder global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -959.640662711 for X = [512, 404.2319].

Figure 39 Two-dimensional Egg Holder function

Function Properties

Variation Bivariate





PAGE 47

Cross-in-Tray B.F

This class defines the Cross-in-Tray global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -2.062611870822739 for x_i = +\- 1.349406608602084 for i=1,2.

Figure 40 Two-dimensional Cross-in-Tray function

Function Properties

Variation Bivariate





PAGE 48

Crowned Cross B.F

This class defines the Crowned Cross global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0.0001. The global minimum is found on the planes x_1 = 0 and x_2 = 0.

Figure 41 Two-dimensional Crowned Cross function

Function Properties

Variation Bivariate





PAGE 49

Deb 1 B.F

This class defines the Deb 1 global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0.0. The number of global minima is 5𝑛 that are evenly spaced in the function landscape, where n represents the dimension of the problem.

Figure 42 Two-dimensional Deb 1 function

Function Properties



Global minimum 0



PAGE 50

Deb 2 B.F

This class defines the Deb 2 global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0.0. The number of global minima is 5𝑛that are evenly spaced in the function landscape, where n represents the dimension of the problem.

Figure 43 Two-dimensional Deb 2 function

Function Properties



Global minimum 0



PAGE 51

DeflectedCorrugatedSpring B.F

This class defines the Deflected Corrugated Spring function global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Where, in this exercise, K = 5 and \alpha = 5.

Here, n represents the number of dimensions and x_i ϵ [0, 2𝛼] for i=1,...,n.

Global optimum: f(x_i) = 0 for x_i = \alpha for i=1,...,n.

Figure 44 Two-dimensional Deflected Corrugated Spring function

Function Properties



Global minimum 0



PAGE 52

Mishra 3 test objective function.

This class defines the Mishra 3 global optimization problem. This is a multimodal minimization problem defined as follows:


Global optimum: for for .

Figure 45Two-dimensional Mishra 3 function

Function Properties

Variation Bivariate


Global minimum 0



PAGE 53

Xin-She Yang 4 test objective function.

This class defines the Xin-She Yang 4 global optimization problem. This is a multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i \in [-10, 10] for i=1,...,n.

Global optimum: f(x_i) = -1 for x_i = 0 for i=1,...,n.

Figure 46 Xin-She Yang 4 test objective function

Function Properties



Global minimum -1



PAGE 54

Valley-Shaped B.F

Wayburn and Seader 1 B.F

This class defines the Wayburn and Seader 1 global optimization problem. This is a

unimodal minimization problem defined as follows:

Here, represents the number of dimensions and for ..

Global optimum: f(x_i) = 0 for x_i = [1, 2].

Figure 47 Two-dimensional Wayburn and Seader 1 function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 55

Rosenbrock’s valley B.F

Rosenbrock’s valley is a classic optimization problem, also known as banana function or the second function of De Jong. The global optimum lays inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial, however convergence to the global optimum is difficult and hence this problem has been frequently used to test the performance of optimization algorithms. Function has the following definition

Test area is usually restricted to hyphercube -2.048≤xi≤2.048, i = 1,…, n. Its global minimum equal f(x) = 0 is obtainable for xi = 0, i = 1,..,n.

Figure 48 Rosenbrock’s valley in 2D,f(x,y) = 100(y ¡ x^2 )^2 + (1- x)^2

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 56

Michalewicz B.F

The Michalewicz function is a continues and multimodal test function (owns n! local optima). The parameter m defines the “steepness” of the valleys or edges. Larger m leads to more difficult search. For very large m the function behaves like a needle in the haystack (the function values for points in the space outside the narrow peaks give very little information on the location of the global optimum). Function has the following definition

It is usually set m=10.Test area is usually restricted to hyphercube 0 ≤xi≤ 𝜋, i = 1,…, n The global minimum value has been approximated by f(x) = -4.687 for n = 5 and by f(x) = -9.66 for n = 10. Respective optimal solutions are not given.

Figure 49

Function Properties


Shape Valley-Shaped

Global minimum Depends on n



PAGE 57

Six-hump camel back B.F

The Six-hump camel back function is a global optimization test function. Within the bounded region it owns six local minima, two of them are global ones. Function has only two variables and the following definition

Test area is usually restricted to hyphercube -3 ≤x1≤ 3, -2 ≤x2≤ 2, i = 1,…, n. Two global minima equal f(x) = -1.0316 are located at (x1, x2) = (-0.0898, 0.7126) and (0.0898,¡0.7126).

Figure 50 Six-hump camel back function

Function Properties

Variation Bivariate

Shape Valley-Shaped




PAGE 58

Deceptive B.F

A deceptive problem is a class of problems in which the total size of the basins for local optimum solutions is much larger than the basin size of the global optimum solution. Clearly, this is a continues and multimodal function. The general form of deceptive function is given by the following formulae

where 𝛽 is an fixed non-linearity factor. It has been defined in the literature at least three types of deceptive problems, depending the form of gi(xi). A complex deceptive problem (Type III), in which

the global optimum is located at xi = 𝛼i, where 𝛼i is a unique random number between 0 and 1 depending on the dimension i. To this aim the following form of auxiliary functions has been proposed

The two other types of deceptive problems (Types I and II) are special cases of the

complex deceptive problem, with 𝛼i = 1 (Type I), or αi = 0 or 1 at random (Type II) for each dimension i, i = 0, . . . , n. Clearly formulae (22) should be suitable adjusted for type I and II.

For all three types of gi(xi), the region with local optima is 5n¡1 times larger than the region with a global optimum in the n-dimensional space. The number of local optima

is 2𝑛 - 1 for Type I and Type II deceptive problems and 3𝑛-1 for Type III.

Function Properties


Shape Valley-Shaped

Global minimum -1



PAGE 59

Figure 51 Deceptive function of Type III in 2D.α1 = 0.3,α2 = 0.7,β= 0.2

42

Figure 52 Deceptive function of Type III in 2D.α1 = 0.3,α2 = 0.7,β = 2.5



PAGE 60

AMGM B.F

This class defines the Arithmetic Mean - Geometric Mean Equality global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for x_1 = x_2 = ... = x_n for i=1,...,n.

Function Properties


Shape Valley-Shaped

Global minimum 0

Figure 53 Two-dimensional Arithmetic Mean - Geometric Mean Equality



PAGE 61

Leon B.F

This class defines the Leon global optimization problem. This is a continues and



Global optimum: f(x_i) = 0 for x_i = 1 for i=1,2.

Figure 54 Two-dimensional Leon function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



PAGE 62

Goldstein-Price B.F

This class defines the Goldstein-Price global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 3 for X = [0, -1].

Figure 55 Two-dimensional Goldstein-Price function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 3



PAGE 63

Judge B.F

This class defines the Judge global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Where, in this exercise:



Figure 56 Two-dimensional Judge function

Function Properties

Variation Bivariate

Shape Valley-Shaped




PAGE 64

Bartels-Conn B.F

This class defines the Bartels-Conn global optimization problem. This is a continues and multimodal minimization problem defined as follows:



Figure 57 4Two-dimensional Bartels-Conn function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 1



PAGE 65

El-Attar-Vidyasagar-Dutta B.F

This class defines the El-Attar-Vidyasagar-Dutta function global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 1.712780354 for X = [3.40918683, -2.17143304].

Figure 58 Two-dimensional El-Attar-Vidyasagar-Dutta function

Function Properties

Variation Bivariate

Shape Valley-Shaped




PAGE 66

Beale B.F

This class defines the Beale global optimization problem. This is a continues and



Global optimum: f(x_i) = 0 for x = [3, 0.5].

Figure 59 Two-dimensional Beale function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



PAGE 67

Bohachevsky B.F

This class defines the Bohachevsky global optimization problem. This is a multimodal




Figure 60 Two-dimensional Bohachevsky Test Objective Function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 68

Brown Test Objective Function

This class defines the Brown global optimization problem. This is a continues and multimodal minimization problem defined as follows:



Figure 5 Two-dimensional Brown function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 69

Bukin 2 B.F

This class defines the Bukin 2 global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_1 ϵ [-15, -5], x_2 ϵ [-3, 3].

Global optimum: f(x_i) = 0 for X = [-10, 0].

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0

Figure 6 Two-dimensional Bukin 2 function



PAGE 70

Bukin 4 B.F




Figure 7Two-dimensional Bukin 4 function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



PAGE 71

Deckkers-Aarts B.F

This class defines the Deckkers-Aarts global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -24777 for X = [0, +\- 15].

Figure 8 Two-dimensional Deckkers-Aarts function

Function Properties

Variation Bivariate

Shape Drop-Shaped

Global minimum -24777



PAGE 72

Bukin 6 B.F




Figure 9Two-dimensional Bukin 6 function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



PAGE 73

Cigar B.F.

This class defines the Cigar global optimization problem. This is a continues and multimodal minimization problem defined as follows:



Figure 10Two-dimensional Cigar function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 74

Csendes B.F

This class defines the Csendes global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0.0 for x_i = 0 for i=1,...,N.

Figure 11 Two-dimensional Csendes function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 75

HimmelBlau B.F

This class defines the HimmelBlau global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for X = [0, 0].

Figure 12 Two-dimensional HimmelBlau function

Function Properties

Variation Bivariate

Shape Multimodal / Valley -Shaped

Global minimum 0



PAGE 76

Decanomial B.F

This class defines the Decanomial function global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for X = [2, -3].

Figure 13 Two-dimensional Decanomial function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 0



PAGE 77

Deckkers-Aarts B.F

This class defines the Deckkers-Aarts global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -24777 for X = [0, +\- 15].

Figure 14 Two-dimensional Deckkers-Aarts function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum -24777



PAGE 78

Dixon and Price B.F

This class defines the Dixon and Price global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Figure 15 Two-dimensional Dixon and Price function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 79

McCormick test objective function.

This class defines the McCormick global optimization problem. This is a multimodal minimization problem defined as follows:

Here, represents the number of dimensions and , ..

Global optimum: for

Figure 16 Two-dimensional McCormick function

Function Properties

Variation Bivariate

Shape Valley-Shaped




PAGE 80

Six Hump Camel test objective function.

This class defines the Six Hump Camel global optimization problem. This is a multimodal minimization problem defined as follows

Here, n represents the number of dimensions and x_i ∈ [-5, 5] for i=1,2.

Global optimum: f(x_i) = -1.031628453489877 for x_i = [0.08984201368301331 , -0.7126564032704135] or x_i [-0.08984201368301331, 0.7126564032704135]

Figure 73 Two-dimensional Six Hump Camel function

Function Properties

Variation Bivariate

Shape Valley/Multimodal -Shaped




PAGE 81

Steep Ridges/Drops-Shaped B.F

Ackley’s B.F

Ackley’s is a widely used multimodal test function. It has the following definition

It is recommended to set a = 20, b = 0.2, c = 2𝜋. Test area is usually restricted to hyphercube -32.768≤xi≤ 32.768, i = 1,…, n. Its global minimum equal f(x)=0 is obtainable for xi = 0, i = 1,..,n.

Figure 7417 An overview of Ackley’s function in 2D,f(x,y)=-xsin(√(|x| )-ysin√(|y| )

Function Properties


Shape Drop-Shaped

Global minimum 0



PAGE 82

Easom B.F

The Easom function is a unimodal test function, where the global minimum has a small area relative to the search space. The function was inverted for minimization. It has only two variables and the following definition

Test area is usually restricted to hyphercube -100 ≤x1≤ 100, -100 ≤x2≤ 100, i = 1,…, n. Its global minimum equal f(x)=-1 is obtainable for (x1,x2)=(𝜋, 𝜋).

Figure 7518 Easom’s function

Function Properties

Variation Bivariate

Shape Drop-Shaped

Global minimum -1



PAGE 83

DropWave B.F

This class defines the DropWave global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_i ϵ [-5.12, 5.12] for i=1,2.

Global optimum: f(x_i) = -1 for x_i = 0 for i=1,2.

Figure 7619 Two-dimensional DropWave function

Function Properties


Shape Drop-Shaped

Global minimum -1



PAGE 84

Shekel105 B.F

This is a continues and multimodal test function. It has the following definition

In this exercise:

Here, n represents the number of dimensions and x_i ϵ [0, 10] for i=1,...,4.

Global optimum: f(x_i) = -10.1527 for x_i = 4 for i=1,...,4

Figure 77

Function Properties


Shape Drop -Shaped




PAGE 85

Cross-Leg-Table B.F

This class defines the Cross-Leg-Table global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = -1. The global minimum is found on the planes x_1 = 0 and x_2 = 0

.

Figure 78 Two-dimensional Cross-Leg-Table function

Function Properties

Variation Bivariate

Shape Drop- Shaped

Global minimum -1



PAGE 86

Xin-She Yang 3 test objective function.

This class defines the Xin-She Yang 3 global optimization problem. This is a multimodal minimization problem defined as follows:

Where, in this exercise, and .


Global optimum: for for .

Figure 79 Two-dimensional Xin-She Yang 3 function

Function Properties


Shape Drop- Shaped

Global minimum -1



PAGE 87

Plate-Shaped B.F

Branins B.F

The Branin function is a global optimization test function having only two variables. The function has three equal-sized global optima, and has the following definition

Here, n represents the number of dimensions and x_i ∈ [-5, 15] for i=1,2.

Global optimum: f(x_i) = 5.559037 for x_i = [-3.2, 12.53].

-

Figure 80 Branins’s function

Function Properties

Variation Bivariate

Shape Plate-Shaped




PAGE 88

Goldstein-Price B.F

The Goldstein-Price function is a global optimization test function. It has only two variables and the following definition

Test area is usually restricted to hyphercube -2 ≤x1≤ 2, -2 ≤x2≤ 2, i = 1,…, n. Its global minimum equal f(x)=3 is obtainable for (x1,x2)=(0,-1).

Figure 8120 Goldstein-Price’s function

Function Properties

Variation Bivariate

Shape Plate-Shaped

Global minimum 3



PAGE 89

Adjiman B.F

This class defines the Adjiman global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Here, n represents the number of dimensions and x_1 𝜖 [-1, 2] and x_2 ϵ [-1, 1].

Global optimum: f(x_i) = -2.02181 for X = [2, 0.10578]

Function Properties

Variation Bivariate

Shape Plate-Shaped


Figure 82 Two-dimensional Adjiman function



PAGE 90

Brent B.F

This class defines the Brent global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for x = [-10, -10].

Figure 83 Two-dimensional Brent function

Function Properties

Variation Bivariate

Shape Plate-Shaped

Global minimum 0



PAGE 91

Cube B.F

This class defines the Cube global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0.0 for X = [1, 1].

Figure 84 Two-dimensional Cube function

Function Properties

Variation Bivariate

Shape Plate-Shaped

Global minimum 0



PAGE 92

Katsuura B.F

This class defines the Katsuura global optimization problem. This is a continues and multimodal minimization problem defined as follows:

Where, in this exercise, d = 32.

Here, n represents the number of dimensions and 𝑥𝑖ϵ [0, 100] for i=1,...,n.


Figure 85 Two-dimensional Katsuura function

Function Properties

Variation Bivariate

Shape Valley-Shaped

Global minimum 1



PAGE 93

Exp2 B.F

This class defines the Exp2 global optimization problem. This is a continues and multimodal minimization problem defined as follows:


Global optimum: f(x_i) = 0 for x_i = [1, 0.1].

Figure 86 Two-dimensional Exp2 function

Function Properties


Shape Valley-Shaped

Global minimum 0



PAGE 94

Univariate Test Functions

Univariate Problem02 B.F

This class defines the Univariate Problem02 global optimization problem. This is a


Bound constraints: .

Global optimum: f(x)=-1.899599 for x = 5.145735.

FIGURE 87

Function Properties

Variation Univariate

Shape Sinusoid-Shaped




PAGE 95




Bound constraints:

Global optimum: f(x)=-12. 03124 for x = -6.7745761

Figure 88 Univariate Problem03 function

Function Properties



Global minimum -12. 03124



PAGE 96







Function Properties






PAGE 97




Bound constraints: . Global optimum: f(x)=-1.48907 for x = 0.96609.


Function Properties






PAGE 98







Function Properties






PAGE 99






Global optimum: f(x)=-1.6013 for x = 5.19978 .

Function Properties






PAGE 100





Global optimum: f(x)=-14.508 for x = -7.083506 .


Function Properties






PAGE 101







Function Properties






PAGE 102







Function Properties






PAGE 103







Function Properties



Global minimum -1.5



PAGE 104





Global optimum: f(x)=-1 for x = \pi.


Function Properties



Global minimum -1



PAGE 105





Global optimum: f(x)=-1.5874 for x = 1/√2 .


Function Properties






PAGE 106







Function Properties






PAGE 107







Function Properties






PAGE 108





Global optimum: f(x)=0 for x = 2.


Function Properties



Global minimum 0



PAGE 109







Function Properties






PAGE 110







Function Properties






PAGE 111





Global optimum: f(x)=exp −27𝜋

2 - 1 for x = 9 𝜋/2.


Function Properties



Global minimum exp( −27𝜋

2) - 1



PAGE 112

Source Code

MVF Source Code Mvf.c is a library of multidimensional functions written in C for unconstrained global

optimization or with simple box constraints.

( )

{

* ;

} ;

( * )

{

;

= ;

( = ; < ; ++) {

( > ) = ;

}

;

}

( )

{

( < ) ;

;

}

( * * )

{

;

;

= ;

( = ; < ; ++) {

= ( ) ;

+= * ;

}

;

}

( * * )

{

;

;

= ( ) ;



PAGE 113

( = ; < ; ++) {

= ( ) ;

( > ) {

= ;

}

}

;

}

( * )

{

;

;

= ;

( = ; < ; ++) {

*= ;

}

;

}

double mvfAckley(int n, double *x)

*

:

: | _ |



PAGE 114

double mvfBeale(int n, double *x)

*

:

: | _ |

: ( )

*

{

( + * ) +

( + * * ) +

( + * ( ) ) ;

}

double mvfBohachevsky1(int n, double *x)

*

:

: ( )

: | |



PAGE 115

*

{

( + * ) +

( * + ) ;

}

double mvfBoxBetts(int n, double * x)

*

:

:

: ( )

*

{

;

= = = ;

= ;

( = ;



PAGE 116

double mvfBranin2(int n, double *x)

{

( * + ( * _ * ) )+

( ( * _ * ) ) ;

}

double mvfCamel3(int n, double *x)

*

:

: | |



PAGE 117

}

= {

} ;

double mvfCola( int n, double * x )

{

= ;

= ;

= { } ;

( = ; < ; ++)

= ;

( = ; < ; ++)

( = ; < ; ++) {

= ;

( = ; < ; ++ )

+= ( * + * + ) ;

+= ( ( ) ) ;

++;

}

;

}

double mvfColville(int n, double *x)

*

:

: |



PAGE 118

+ * ( ( ) + ( ) )

+ * ( )*( ) ;

}

double mvfCorana(int n, double *x)

*

: | | <

*

{

;

;

;

= { } ;

= ;

( = ; < ; ++) {

= ( ( ) + ) * ( ) * ;

( ( ) < ) {

+= * ( * ( ) ) * ;

} {

+= * * ;

}

} ;

;

}

double mvfEasom(int n, double *x)

*

( )

: | |



PAGE 119

double mvfEggholder(int n, double *x)

*

:

: | _ | <

*

{

;

;

= ;

( = ; < ; ++) {

+= ( + + ) * ( ( ( + + * + ))) +

( ( ( ( + + ))) ) * ( ) ;

}

;

}

double mvfExp2(int n, double *x)

*

:

:



PAGE 120

double mvfFraudensteinRoth(int n, double *x)

*

*

{

( + + (( ) * ) * )

+ ( + + (( + + ) * ) * ) ;

}

( * )

{

;

* == *

= ( ) * ( ) ( ( ) * ( ) ) ;

* ;

}

double mvfGeneralizedRosenbrock(int n, double * x)

{

= ;

;

( = ; < ; ++ )

+= * ( * )

+ ( ) ;

;

}

double mvfGoldsteinPrice(int n, double * x)

*

: =

: | _ |



PAGE 121

{

;

= ;

= ;

( = ; < ; ++) {

+= * ;

*= ( ( ( ) ( + )) ) ;

}

+ ;

}

double mvfHansen(int n, double * x)

{

( ( )+ * ( + )

+ * ( * + )+ * ( * + )

+ * ( * + ))*( ( * + )

+ * ( * + )

+ * ( * + )

+ * ( * + )

+ * ( * + )) ;

}

double mvfHartman3(int n, double *x)

{

;

;

= {

{ }

{ }

{ }

{ }

} ;

= { } ;

= {

{ }

{ }

{ }

{ }

} ;

= ;

= ;

( = ; < ; ++) {



PAGE 122

= ;

( = ; < ; ++) {

= ;

+= * ( * ) ;

}

+= * ( ) ;

}

;

}

double mvfHartman6(int n, double *x)

*

_ ( = )

:



PAGE 123

;

}

double mvfHimmelblau(int n, double *x)

*

:

: | | <

: ( )

*

{

( * + ) + ( + * ) ;

}

double mvfHolzman1(int n, double *x)

*

:

:



PAGE 124

+= * ( ) ;

}

;

}

double mvfHosaki(int n, double *x)

*

=

*

{

( + *

( + *

( + *

( + *

)) ) ) *

* * ( ) ;

}

double mvfHyperellipsoid(int n, double *x)

*

=

: | |



PAGE 125

double mvfKatsuuras(int n, double *x)

*

= : ( )

: | |



PAGE 126

= ;

( = ; < ; ++) {

= * ;

= ( *( + * ) ( + * + )) ;

+= * ;

}

;

}

= {

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

} ;

= {



PAGE 127

} ;

double mvfLangerman(int n, double *x)

{

;

;

= ;

= ;

( = ; < ; ++ ){

= ( ) ;

= * ( ( ) _ ) * ( _ * ) ;

}

;

}

double mvfLennardJones(int n, double *x)

{

;

;

;

= ;

( = ; < ; ++) {

= * ;

( = + ; < ; ++) {

= ;

= = * ;

+= * ;

= + + ;

+= * ;

= + + ;

+= * ;

( < ) { * ? *

;

}

{

= ( * * ) ;

+= ( ) * ;



PAGE 128

}

}

}

;

}

double mvfLeon(int n, double *x)

*

*

{

= * * ;

= ;

* * + * ;

}

double mvfLevy(int n, double * x)

*

= = ( )

= = ( \ )

*

{

;

= ;

( = ;



PAGE 129

: |



PAGE 130

= = ( ) ;

( = ; < ; ++) {

= ( ) ;

+= ;

*= ;

}

+ ;

}

* *

_ = ;

_ = ; * *

( )

{

_ = ;

}

( )

{

_ = ;

}

double mvfNeumaierPerm(int n, double x[])

*

= ( + ) * =

: =

: ( ) = ( ) ( ) ( ^ ) ( ^ )

*

{

;

= ;

( = ; < ; ++) {

( = ; < ; ++) {

+= ( ( + + ) + _ ) * ( ( ( + ) ( + )) ) ;

}

}

;

}

double mvfNeumaierPerm0(int n, double x[])

*



PAGE 131

: _ =

* = = ( + )

( ) = ( ) ( )

*

{

;

= ;

( = ; < ; ++) {

( = ; < ; ++) {

+= (( + )+ _ ) * ( ( + ) ( ( + ) ) ) ;

}

}

;

}

_ _ = { } ;

double mvfNeumaierPowersum(int n, double x[])

*

* =

: _ _

* = ( )

*

{

;

;

= ;

( = ; < ; ++) {

= ;

( = ; < ; ++) {

+= ( + ) ;

}

+= ( _ _ ) ;

}

;

}

double mvfNeumaierTrid(int n, double x[])



PAGE 132

*

^ ^

_ = ( + ) ( )= ( + )( )

*

{

;

;

= ;

( = ; < ; ++) {

+= ( ) ;

( ) += * ;

}

;

}

double mvfOddsquare(int n, double *x)

*

: =

: | |



PAGE 133

+= * + * ;

*= ;

}

( ) ;

}

double mvfPlateau(int n, double *x)

*

: | | < =

:

" "

*

{

;

= ;

( = ; < ; ++) {

+= ( ) ;

}

+ ;

}

double mvfPowell(int n, double *x)

{

;

;

= ;

( = ; < ; ++) {

+= ( * + * * )

+ * ( * * )

+ ( * * * )

+ * ( * * ) ;

}

;

}



PAGE 134

double mvfQuarticNoiseU(int n, double *x)

*

: | _ | <

_ =

*

{

;

;

= ;

( = ; < ; ++) {

= ;

= * ;

+= * + ( ) ;

}

;

}

double mvfQuarticNoiseZ(int n, double *x)

*

| _ | <

_ =

*

{

;

;

= ;

( = ; < ; ++) {

= ;

= * ;

+= ( * * + Z()) ;

}

;

}

double mvfRana(int n, double *x)

*



PAGE 135

= | _ | <

*

{

;

;

= ;

( = ; < ; ++) {

= ( ( + + + )) ;

= ( ( + + )) ;

+= ( + + ) * ( ) * ( ) + ( ) * ( ) * ;

}

;

}

double mvfRastrigin(int n, double *x)

*

= | _ | <

_ =

*

{

;

;

( > ) {

= ;

}

= ;

( = ; < ; ++) {

= ;

+= * ( * _ * ) ;

}

+ * ;

}

double mvfRastrigin2(int n, double *x)

*

=



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