Test functions for optimization

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Short description: Functions used to evaluate optimization algorithms

In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:

  • Convergence rate.
  • Precision.
  • Robustness.
  • General performance.


Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given.

The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck,[1] Haupt et al.[2] and from Rody Oldenhuis software.[3] Given the number of problems (55 in total), just a few are presented here.

The test functions used to evaluate the algorithms for MOP were taken from Deb,[4] Binh et al.[5] and Binh.[6] The software developed by Deb can be downloaded,[7] which implements the NSGA-II procedure with GAs, or the program posted on Internet,[8] which implements the NSGA-II procedure with ES.

Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein.

Test functions for single-objective optimization

Name Plot Formula Global minimum Search domain
Rastrigin function Rastrigin function for n=2 [math]\displaystyle{ f(\mathbf{x}) = A n + \sum_{i=1}^n \left[x_i^2 - A\cos(2 \pi x_i)\right] }[/math]

[math]\displaystyle{ \text{where: } A=10 }[/math]

[math]\displaystyle{ f(0, \dots, 0) = 0 }[/math] [math]\displaystyle{ -5.12\le x_{i} \le 5.12 }[/math]
Ackley function Ackley's function for n=2 [math]\displaystyle{ f(x,y) = -20\exp\left[-0.2\sqrt{0.5\left(x^{2}+y^{2}\right)}\right] }[/math]

[math]\displaystyle{ -\exp\left[0.5\left(\cos 2\pi x + \cos 2\pi y \right)\right] + e + 20 }[/math]

[math]\displaystyle{ f(0,0) = 0 }[/math] [math]\displaystyle{ -5\le x,y \le 5 }[/math]
Sphere function Sphere function for n=2 [math]\displaystyle{ f(\boldsymbol{x}) = \sum_{i=1}^{n} x_{i}^{2} }[/math] [math]\displaystyle{ f(x_{1}, \dots, x_{n}) = f(0, \dots, 0) = 0 }[/math] [math]\displaystyle{ -\infty \le x_{i} \le \infty }[/math], [math]\displaystyle{ 1 \le i \le n }[/math]
Rosenbrock function Rosenbrock's function for n=2 [math]\displaystyle{ f(\boldsymbol{x}) = \sum_{i=1}^{n-1} \left[ 100 \left(x_{i+1} - x_{i}^{2}\right)^{2} + \left(1 - x_{i}\right)^{2}\right] }[/math] [math]\displaystyle{ \text{Min} = \begin{cases} n=2 & \rightarrow \quad f(1,1) = 0, \\ n=3 & \rightarrow \quad f(1,1,1) = 0, \\ n\gt 3 & \rightarrow \quad f(\underbrace{1,\dots,1}_{n \text{ times}}) = 0 \\ \end{cases} }[/math] [math]\displaystyle{ -\infty \le x_{i} \le \infty }[/math], [math]\displaystyle{ 1 \le i \le n }[/math]
Beale function Beale's function [math]\displaystyle{ f(x,y) = \left( 1.5 - x + xy \right)^{2} + \left( 2.25 - x + xy^{2}\right)^{2} }[/math]

[math]\displaystyle{ + \left(2.625 - x+ xy^{3}\right)^{2} }[/math]

[math]\displaystyle{ f(3, 0.5) = 0 }[/math] [math]\displaystyle{ -4.5 \le x,y \le 4.5 }[/math]
Goldstein–Price function Goldstein–Price function [math]\displaystyle{ f(x,y) = \left[1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right] }[/math]

[math]\displaystyle{ \left[30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right] }[/math]

[math]\displaystyle{ f(0, -1) = 3 }[/math] [math]\displaystyle{ -2 \le x,y \le 2 }[/math]
Booth function Booth's function [math]\displaystyle{ f(x,y) = \left( x + 2y -7\right)^{2} + \left(2x +y - 5\right)^{2} }[/math] [math]\displaystyle{ f(1,3) = 0 }[/math] [math]\displaystyle{ -10 \le x,y \le 10 }[/math]
Bukin function N.6 Bukin function N.6 [math]\displaystyle{ f(x,y) = 100\sqrt{\left|y - 0.01x^{2}\right|} + 0.01 \left|x+10 \right|.\quad }[/math] [math]\displaystyle{ f(-10,1) = 0 }[/math] [math]\displaystyle{ -15\le x \le -5 }[/math], [math]\displaystyle{ -3\le y \le 3 }[/math]
Matyas function Matyas function [math]\displaystyle{ f(x,y) = 0.26 \left( x^{2} + y^{2}\right) - 0.48 xy }[/math] [math]\displaystyle{ f(0,0) = 0 }[/math] [math]\displaystyle{ -10\le x,y \le 10 }[/math]
Lévi function N.13 Lévi function N.13 [math]\displaystyle{ f(x,y) = \sin^{2} 3\pi x + \left(x-1\right)^{2}\left(1+\sin^{2} 3\pi y\right) }[/math]

[math]\displaystyle{ +\left(y-1\right)^{2}\left(1+\sin^{2} 2\pi y\right) }[/math]

[math]\displaystyle{ f(1,1) = 0 }[/math] [math]\displaystyle{ -10\le x,y \le 10 }[/math]
Himmelblau's function Himmelblau's function [math]\displaystyle{ f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2.\quad }[/math] [math]\displaystyle{ \text{Min} = \begin{cases} f\left(3.0, 2.0\right) & = 0.0 \\ f\left(-2.805118, 3.131312\right) & = 0.0 \\ f\left(-3.779310, -3.283186\right) & = 0.0 \\ f\left(3.584428, -1.848126\right) & = 0.0 \\ \end{cases} }[/math] [math]\displaystyle{ -5\le x,y \le 5 }[/math]
Three-hump camel function Three Hump Camel function [math]\displaystyle{ f(x,y) = 2x^{2} - 1.05x^{4} + \frac{x^{6}}{6} + xy + y^{2} }[/math] [math]\displaystyle{ f(0,0) = 0 }[/math] [math]\displaystyle{ -5\le x,y \le 5 }[/math]
Easom function Easom function [math]\displaystyle{ f(x,y) = -\cos \left(x\right)\cos \left(y\right) \exp\left(-\left(\left(x-\pi\right)^{2} + \left(y-\pi\right)^{2}\right)\right) }[/math] [math]\displaystyle{ f(\pi , \pi) = -1 }[/math] [math]\displaystyle{ -100\le x,y \le 100 }[/math]
Cross-in-tray function Cross-in-tray function [math]\displaystyle{ f(x,y) = -0.0001 \left[ \left| \sin x \sin y \exp \left(\left|100 - \frac{\sqrt{x^{2} + y^{2}}}{\pi} \right|\right)\right| + 1 \right]^{0.1} }[/math] [math]\displaystyle{ \text{Min} = \begin{cases} f\left(1.34941, -1.34941\right) & = -2.06261 \\ f\left(1.34941, 1.34941\right) & = -2.06261 \\ f\left(-1.34941, 1.34941\right) & = -2.06261 \\ f\left(-1.34941,-1.34941\right) & = -2.06261 \\ \end{cases} }[/math] [math]\displaystyle{ -10\le x,y \le 10 }[/math]
Eggholder function[9][10] Eggholder function [math]\displaystyle{ f(x,y) = - \left(y+47\right) \sin \sqrt{\left|\frac{x}{2}+\left(y+47\right)\right|} - x \sin \sqrt{\left|x - \left(y + 47 \right)\right|} }[/math] [math]\displaystyle{ f(512, 404.2319) = -959.6407 }[/math] [math]\displaystyle{ -512\le x,y \le 512 }[/math]
Hölder table function Holder table function [math]\displaystyle{ f(x,y) = - \left|\sin x \cos y \exp \left(\left|1 - \frac{\sqrt{x^{2} + y^{2}}}{\pi} \right|\right)\right| }[/math] [math]\displaystyle{ \text{Min} = \begin{cases} f\left(8.05502, 9.66459\right) & = -19.2085 \\ f\left(-8.05502, 9.66459\right) & = -19.2085 \\ f\left(8.05502,-9.66459\right) & = -19.2085 \\ f\left(-8.05502,-9.66459\right) & = -19.2085 \end{cases} }[/math] [math]\displaystyle{ -10\le x,y \le 10 }[/math]
McCormick function McCormick function [math]\displaystyle{ f(x,y) = \sin \left(x+y\right) + \left(x-y\right)^{2} - 1.5x + 2.5y + 1 }[/math] [math]\displaystyle{ f(-0.54719,-1.54719) = -1.9133 }[/math] [math]\displaystyle{ -1.5\le x \le 4 }[/math], [math]\displaystyle{ -3\le y \le 4 }[/math]
Schaffer function N. 2 Schaffer function N.2 [math]\displaystyle{ f(x,y) = 0.5 + \frac{\sin^{2}\left(x^{2} - y^{2}\right) - 0.5}{\left[1 + 0.001\left(x^{2} + y^{2}\right) \right]^{2}} }[/math] [math]\displaystyle{ f(0, 0) = 0 }[/math] [math]\displaystyle{ -100\le x,y \le 100 }[/math]
Schaffer function N. 4 Schaffer function N.4 [math]\displaystyle{ f(x,y) = 0.5 + \frac{\cos^{2}\left[\sin \left( \left|x^{2} - y^{2}\right|\right)\right] - 0.5}{\left[1 + 0.001\left(x^{2} + y^{2}\right) \right]^{2}} }[/math] [math]\displaystyle{ \text{Min} = \begin{cases} f\left(0,1.25313\right) & = 0.292579 \\ f\left(0,-1.25313\right) & = 0.292579 \\ f\left(1.25313,0\right) & = 0.292579 \\ f\left(-1.25313,0\right) & = 0.292579 \end{cases} }[/math] [math]\displaystyle{ -100\le x,y \le 100 }[/math]
Styblinski–Tang function Styblinski-Tang function [math]\displaystyle{ f(\boldsymbol{x}) = \frac{\sum_{i=1}^{n} x_{i}^{4} - 16x_{i}^{2} + 5x_{i}}{2} }[/math] [math]\displaystyle{ -39.16617n \lt f(\underbrace{-2.903534, \ldots, -2.903534}_{n \text{ times}} ) \lt -39.16616n }[/math] [math]\displaystyle{ -5\le x_{i} \le 5 }[/math], [math]\displaystyle{ 1\le i \le n }[/math]..
Shekel function A Shekel function in 2 dimensions and with 10 maxima [math]\displaystyle{ f(\vec{x}) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ji})^2 \right)^{-1} }[/math]

or, similarly, [math]\displaystyle{ f(x_1,x_2,...,x_{n-1},x_n) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ij})^2 \right)^{-1} }[/math]

[math]\displaystyle{ -\infty \le x_{i} \le \infty }[/math], [math]\displaystyle{ 1 \le i \le n }[/math]

Test functions for constrained optimization

Name Plot Formula Global minimum Search domain
Rosenbrock function constrained with a cubic and a line[11] Rosenbrock function constrained with a cubic and a line [math]\displaystyle{ f(x,y) = (1-x)^2 + 100(y-x^2)^2 }[/math],

subjected to: [math]\displaystyle{ (x-1)^3 - y + 1 \le 0 \text{ and } x + y - 2 \le 0 }[/math]

[math]\displaystyle{ f(1.0,1.0) = 0 }[/math] [math]\displaystyle{ -1.5\le x \le 1.5 }[/math], [math]\displaystyle{ -0.5\le y \le 2.5 }[/math]
Rosenbrock function constrained to a disk[12] Rosenbrock function constrained to a disk [math]\displaystyle{ f(x,y) = (1-x)^2 + 100(y-x^2)^2 }[/math],

subjected to: [math]\displaystyle{ x^2 + y^2 \le 2 }[/math]

[math]\displaystyle{ f(1.0,1.0) = 0 }[/math] [math]\displaystyle{ -1.5\le x \le 1.5 }[/math], [math]\displaystyle{ -1.5\le y \le 1.5 }[/math]
Mishra's Bird function - constrained[13][14] Bird function (constrained) [math]\displaystyle{ f(x,y) = \sin(y) e^{\left [(1-\cos x)^2\right]} + \cos(x) e^{\left [(1-\sin y)^2 \right]} + (x-y)^2 }[/math],

subjected to: [math]\displaystyle{ (x+5)^2 + (y+5)^2 \lt 25 }[/math]

[math]\displaystyle{ f(-3.1302468,-1.5821422) = -106.7645367 }[/math] [math]\displaystyle{ -10\le x \le 0 }[/math], [math]\displaystyle{ -6.5\le y \le 0 }[/math]
Townsend function (modified)[15] Heart constrained multimodal function [math]\displaystyle{ f(x,y) = -[\cos((x-0.1)y)]^2 - x \sin(3x+y) }[/math],

subjected to:[math]\displaystyle{ x^2+y^2 \lt \left[2\cos t - \frac 1 2 \cos 2t - \frac 1 4 \cos 3t - \frac 1 8 \cos 4t\right]^2 + [2\sin t]^2 }[/math] where: t = Atan2(x,y)

[math]\displaystyle{ f(2.0052938,1.1944509) = -2.0239884 }[/math] [math]\displaystyle{ -2.25\le x \le 2.25 }[/math], [math]\displaystyle{ -2.5\le y \le 1.75 }[/math]
Gomez and Levy function (modified)[16] Gomez and Levy Function [math]\displaystyle{ f(x,y) = 4 x^2 - 2.1 x^4 + \frac 1 3 x^6 + xy - 4y^2 +4 y^4 }[/math],

subjected to:[math]\displaystyle{ -\sin(4 \pi x) + 2\sin^2(2 \pi y) \le 1.5 }[/math]

[math]\displaystyle{ f(0.08984201,-0.7126564) = -1.031628453 }[/math] [math]\displaystyle{ -1\le x \le 0.75 }[/math], [math]\displaystyle{ -1\le y \le 1 }[/math]
Simionescu function[17] Simionescu function [math]\displaystyle{ f(x,y) = 0.1xy }[/math],

subjected to: [math]\displaystyle{ x^2+y^2\le\left[r_{T}+r_{S}\cos\left(n \arctan \frac{x}{y} \right)\right]^2 }[/math] [math]\displaystyle{ \text{where: } r_{T}=1, r_{S}=0.2 \text{ and } n = 8 }[/math]

[math]\displaystyle{ f(\pm 0.84852813,\mp 0.84852813) = -0.072 }[/math] [math]\displaystyle{ -1.25\le x,y \le 1.25 }[/math]

Test functions for multi-objective optimization

Name Plot Functions Constraints Search domain
Binh and Korn function:[5] Binh and Korn function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = 4x^{2} + 4y^{2} \\ f_{2}\left(x,y\right) = \left(x - 5\right)^{2} + \left(y - 5\right)^{2} \\ \end{cases} }[/math] [math]\displaystyle{ \text{s.t.} = \begin{cases} g_{1}\left(x,y\right) = \left(x - 5\right)^{2} + y^{2} \leq 25 \\ g_{2}\left(x,y\right) = \left(x - 8\right)^{2} + \left(y + 3\right)^{2} \geq 7.7 \\ \end{cases} }[/math] [math]\displaystyle{ 0\le x \le 5 }[/math], [math]\displaystyle{ 0\le y \le 3 }[/math]
Chankong and Haimes function:[18] Chakong and Haimes function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = 2 + \left(x-2\right)^{2} + \left(y-1\right)^{2} \\ f_{2}\left(x,y\right) = 9x - \left(y - 1\right)^{2} \\ \end{cases} }[/math] [math]\displaystyle{ \text{s.t.} = \begin{cases} g_{1}\left(x,y\right) = x^{2} + y^{2} \leq 225 \\ g_{2}\left(x,y\right) = x - 3y + 10 \leq 0 \\ \end{cases} }[/math] [math]\displaystyle{ -20\le x,y \le 20 }[/math]
Fonseca–Fleming function:[19] Fonseca and Fleming function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = 1 - \exp \left[-\sum_{i=1}^{n} \left(x_{i} - \frac{1}{\sqrt{n}} \right)^{2} \right] \\ f_{2}\left(\boldsymbol{x}\right) = 1 - \exp \left[-\sum_{i=1}^{n} \left(x_{i} + \frac{1}{\sqrt{n}} \right)^{2} \right] \\ \end{cases} }[/math] [math]\displaystyle{ -4\le x_{i} \le 4 }[/math], [math]\displaystyle{ 1\le i \le n }[/math]
Test function 4:[6] Test function 4.[6] [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = x^{2} - y \\ f_{2}\left(x,y\right) = -0.5x - y - 1 \\ \end{cases} }[/math] [math]\displaystyle{ \text{s.t.} = \begin{cases} g_{1}\left(x,y\right) = 6.5 - \frac{x}{6} - y \geq 0 \\ g_{2}\left(x,y\right) = 7.5 - 0.5x - y \geq 0 \\ g_{3}\left(x,y\right) = 30 - 5x - y \geq 0 \\ \end{cases} }[/math] [math]\displaystyle{ -7\le x,y \le 4 }[/math]
Kursawe function:[20] Kursawe function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = \sum_{i=1}^{2} \left[-10 \exp \left(-0.2 \sqrt{x_{i}^{2} + x_{i+1}^{2}} \right) \right] \\ & \\ f_{2}\left(\boldsymbol{x}\right) = \sum_{i=1}^{3} \left[\left|x_{i}\right|^{0.8} + 5 \sin \left(x_{i}^{3} \right) \right] \\ \end{cases} }[/math] [math]\displaystyle{ -5\le x_{i} \le 5 }[/math], [math]\displaystyle{ 1\le i \le 3 }[/math].
Schaffer function N. 1:[21] Schaffer function N.1 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x\right) = x^{2} \\ f_{2}\left(x\right) = \left(x-2\right)^{2} \\ \end{cases} }[/math] [math]\displaystyle{ -A\le x \le A }[/math]. Values of [math]\displaystyle{ A }[/math] from [math]\displaystyle{ 10 }[/math] to [math]\displaystyle{ 10^{5} }[/math] have been used successfully. Higher values of [math]\displaystyle{ A }[/math] increase the difficulty of the problem.
Schaffer function N. 2: Schaffer function N.2 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x\right) = \begin{cases} -x, & \text{if } x \le 1 \\ x-2, & \text{if } 1 \lt x \le 3 \\ 4-x, & \text{if } 3 \lt x \le 4 \\ x-4, & \text{if } x \gt 4 \\ \end{cases} \\ f_{2}\left(x\right) = \left(x-5\right)^{2} \\ \end{cases} }[/math] [math]\displaystyle{ -5\le x \le 10 }[/math].
Poloni's two objective function: Poloni's two objective function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = \left[1 + \left(A_{1} - B_{1}\left(x,y\right) \right)^{2} + \left(A_{2} - B_{2}\left(x,y\right) \right)^{2} \right] \\ f_{2}\left(x,y\right) = \left(x + 3\right)^{2} + \left(y + 1 \right)^{2} \\ \end{cases} }[/math]

[math]\displaystyle{ \text{where} = \begin{cases} A_{1} = 0.5 \sin \left(1\right) - 2 \cos \left(1\right) + \sin \left(2\right) - 1.5 \cos \left(2\right) \\ A_{2} = 1.5 \sin \left(1\right) - \cos \left(1\right) + 2 \sin \left(2\right) - 0.5 \cos \left(2\right) \\ B_{1}\left(x,y\right) = 0.5 \sin \left(x\right) - 2 \cos \left(x\right) + \sin \left(y\right) - 1.5 \cos \left(y\right) \\ B_{2}\left(x,y\right) = 1.5 \sin \left(x\right) - \cos \left(x\right) + 2 \sin \left(y\right) - 0.5 \cos \left(y\right) \end{cases} }[/math]

[math]\displaystyle{ -\pi\le x,y \le \pi }[/math]
Zitzler–Deb–Thiele's function N. 1:[22] Zitzler-Deb-Thiele's function N.1 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = x_{1} \\ f_{2}\left(\boldsymbol{x}\right) = g\left(\boldsymbol{x}\right) h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) \\ g\left(\boldsymbol{x}\right) = 1 + \frac{9}{29} \sum_{i=2}^{30} x_{i} \\ h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) = 1 - \sqrt{\frac{f_{1}\left(\boldsymbol{x}\right)}{g\left(\boldsymbol{x}\right)}} \\ \end{cases} }[/math] [math]\displaystyle{ 0\le x_{i} \le 1 }[/math], [math]\displaystyle{ 1\le i \le 30 }[/math].
Zitzler–Deb–Thiele's function N. 2:[22] Zitzler-Deb-Thiele's function N.2 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = x_{1} \\ f_{2}\left(\boldsymbol{x}\right) = g\left(\boldsymbol{x}\right) h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) \\ g\left(\boldsymbol{x}\right) = 1 + \frac{9}{29} \sum_{i=2}^{30} x_{i} \\ h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) = 1 - \left(\frac{f_{1}\left(\boldsymbol{x}\right)}{g\left(\boldsymbol{x}\right)}\right)^{2} \\ \end{cases} }[/math] [math]\displaystyle{ 0\le x_{i} \le 1 }[/math], [math]\displaystyle{ 1\le i \le 30 }[/math].
Zitzler–Deb–Thiele's function N. 3:[22] Zitzler-Deb-Thiele's function N.3 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = x_{1} \\ f_{2}\left(\boldsymbol{x}\right) = g\left(\boldsymbol{x}\right) h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) \\ g\left(\boldsymbol{x}\right) = 1 + \frac{9}{29} \sum_{i=2}^{30} x_{i} \\ h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) = 1 - \sqrt{\frac{f_{1}\left(\boldsymbol{x}\right)}{g\left(\boldsymbol{x} \right)}} - \left(\frac{f_{1}\left(\boldsymbol{x}\right)}{g\left(\boldsymbol{x}\right)} \right) \sin \left(10 \pi f_{1} \left(\boldsymbol{x} \right) \right) \end{cases} }[/math] [math]\displaystyle{ 0\le x_{i} \le 1 }[/math], [math]\displaystyle{ 1\le i \le 30 }[/math].
Zitzler–Deb–Thiele's function N. 4:[22] Zitzler-Deb-Thiele's function N.4 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = x_{1} \\ f_{2}\left(\boldsymbol{x}\right) = g\left(\boldsymbol{x}\right) h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) \\ g\left(\boldsymbol{x}\right) = 91 + \sum_{i=2}^{10} \left(x_{i}^{2} - 10 \cos \left(4 \pi x_{i}\right) \right) \\ h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) = 1 - \sqrt{\frac{f_{1}\left(\boldsymbol{x}\right)}{g\left(\boldsymbol{x} \right)}} \end{cases} }[/math] [math]\displaystyle{ 0\le x_{1} \le 1 }[/math], [math]\displaystyle{ -5\le x_{i} \le 5 }[/math], [math]\displaystyle{ 2\le i \le 10 }[/math]
Zitzler–Deb–Thiele's function N. 6:[22] Zitzler-Deb-Thiele's function N.6 [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = 1 - \exp \left(-4x_{1}\right)\sin^{6}\left(6 \pi x_{1} \right) \\ f_{2}\left(\boldsymbol{x}\right) = g\left(\boldsymbol{x}\right) h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) \\ g\left(\boldsymbol{x}\right) = 1 + 9 \left[\frac{\sum_{i=2}^{10} x_{i}}{9}\right]^{0.25} \\ h \left(f_{1}\left(\boldsymbol{x}\right),g\left(\boldsymbol{x}\right)\right) = 1 - \left(\frac{f_{1}\left(\boldsymbol{x}\right)}{g\left(\boldsymbol{x} \right)}\right)^{2} \\ \end{cases} }[/math] [math]\displaystyle{ 0\le x_{i} \le 1 }[/math], [math]\displaystyle{ 1\le i \le 10 }[/math].
Osyczka and Kundu function:[23] Osyczka and Kundu function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(\boldsymbol{x}\right) = -25 \left(x_{1}-2\right)^{2} - \left(x_{2}-2\right)^{2} - \left(x_{3}-1\right)^{2} - \left(x_{4}-4\right)^{2} - \left(x_{5}-1\right)^{2} \\ f_{2}\left(\boldsymbol{x}\right) = \sum_{i=1}^{6} x_{i}^{2} \\ \end{cases} }[/math] [math]\displaystyle{ \text{s.t.} = \begin{cases} g_{1}\left(\boldsymbol{x}\right) = x_{1} + x_{2} - 2 \geq 0 \\ g_{2}\left(\boldsymbol{x}\right) = 6 - x_{1} - x_{2} \geq 0 \\ g_{3}\left(\boldsymbol{x}\right) = 2 - x_{2} + x_{1} \geq 0 \\ g_{4}\left(\boldsymbol{x}\right) = 2 - x_{1} + 3x_{2} \geq 0 \\ g_{5}\left(\boldsymbol{x}\right) = 4 - \left(x_{3}-3\right)^{2} - x_{4} \geq 0 \\ g_{6}\left(\boldsymbol{x}\right) = \left(x_{5} - 3\right)^{2} + x_{6} - 4 \geq 0 \end{cases} }[/math] [math]\displaystyle{ 0\le x_{1},x_{2},x_{6} \le 10 }[/math], [math]\displaystyle{ 1\le x_{3},x_{5} \le 5 }[/math], [math]\displaystyle{ 0\le x_{4} \le 6 }[/math].
CTP1 function (2 variables):[4][24] CTP1 function (2 variables).[4] [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = x \\ f_{2}\left(x,y\right) = \left(1 + y\right) \exp \left(-\frac{x}{1+y} \right) \end{cases} }[/math] [math]\displaystyle{ \text{s.t.} = \begin{cases} g_{1}\left(x,y\right) = \frac{f_{2}\left(x,y\right)}{0.858 \exp \left(-0.541 f_{1}\left(x,y\right)\right)} \geq 1 \\ g_{2}\left(x,y\right) = \frac{f_{2}\left(x,y\right)}{0.728 \exp \left(-0.295 f_{1}\left(x,y\right)\right)} \geq 1 \end{cases} }[/math] [math]\displaystyle{ 0\le x,y \le 1 }[/math].
Constr-Ex problem:[4] Constr-Ex problem.[4] [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = x \\ f_{2}\left(x,y\right) = \frac{1 + y}{x} \\ \end{cases} }[/math] [math]\displaystyle{ \text{s.t.} = \begin{cases} g_{1}\left(x,y\right) = y + 9x \geq 6 \\ g_{2}\left(x,y\right) = -y + 9x \geq 1 \\ \end{cases} }[/math] [math]\displaystyle{ 0.1\le x \le 1 }[/math], [math]\displaystyle{ 0\le y \le 5 }[/math]
Viennet function: Viennet function [math]\displaystyle{ \text{Minimize} = \begin{cases} f_{1}\left(x,y\right) = 0.5\left(x^{2} + y^{2}\right) + \sin\left(x^{2} + y^{2} \right) \\ f_{2}\left(x,y\right) = \frac{\left(3x - 2y + 4\right)^{2}}{8} + \frac{\left(x - y + 1\right)^{2}}{27} + 15 \\ f_{3}\left(x,y\right) = \frac{1}{x^{2} + y^{2} + 1} - 1.1 \exp \left(- \left(x^{2} + y^{2} \right) \right) \\ \end{cases} }[/math] [math]\displaystyle{ -3\le x,y \le 3 }[/math].

References

  1. ↑ Bäck, Thomas (1995). Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms. Oxford: Oxford University Press. p. 328. ISBN 978-0-19-509971-3. 
  2. ↑ Haupt, Randy L. Haupt, Sue Ellen (2004). Practical genetic algorithms with CD-Rom (2nd ed.). New York: J. Wiley. ISBN 978-0-471-45565-3. 
  3. ↑ Oldenhuis, Rody. "Many test functions for global optimizers". Mathworks. http://www.mathworks.com/matlabcentral/fileexchange/23147-many-testfunctions-for-global-optimizers. 
  4. ↑ 4.0 4.1 4.2 4.3 4.4 Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. ISBN 0-471-87339-X.
  5. ↑ 5.0 5.1 Binh T. and Korn U. (1997) MOBES: A Multiobjective Evolution Strategy for Constrained Optimization Problems. In: Proceedings of the Third International Conference on Genetic Algorithms. Czech Republic. pp. 176–182
  6. ↑ 6.0 6.1 6.2 Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany
  7. ↑ Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml
  8. ↑ Ortiz, Gilberto A.. "Multi-objective optimization using ES as Evolutionary Algorithm.". Mathworks. http://www.mathworks.com/matlabcentral/fileexchange/35824-multi-objective-optimization-using-evolution-strategies-es-as-evolutionary-algorithm-ea. 
  9. ↑ Whitley, Darrell; Rana, Soraya; Dzubera, John; Mathias, Keith E. (1996). "Evaluating evolutionary algorithms". Artificial Intelligence (Elsevier BV) 85 (1–2): 264. doi:10.1016/0004-3702(95)00124-7. ISSN 0004-3702. 
  10. ↑ Vanaret C. (2015) Hybridization of interval methods and evolutionary algorithms for solving difficult optimization problems. PhD thesis. Ecole Nationale de l'Aviation Civile. Institut National Polytechnique de Toulouse, France.
  11. ↑ Simionescu, P.A.; Beale, D. (September 29 – October 2, 2002). "New Concepts in Graphic Visualization of Objective Functions". ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Montreal, Canada. pp. 891–897. http://faculty.tamucc.edu/psimionescu/PDFs/DETC02-DAC-34129.pdf. Retrieved 7 January 2017. 
  12. ↑ "Solve a Constrained Nonlinear Problem - MATLAB & Simulink". https://www.mathworks.com/help/optim/ug/example-nonlinear-constrained-minimization.html?requestedDomain=www.mathworks.com. 
  13. ↑ "Bird Problem (Constrained) | Phoenix Integration". http://www.phoenix-int.com/software/benchmark_report/bird_constrained.php. 
  14. ↑ Mishra, Sudhanshu (2006). "Some new test functions for global optimization and performance of repulsive particle swarm method". MPRA Paper. https://mpra.ub.uni-muenchen.de/2718/. 
  15. ↑ Townsend, Alex (January 2014). "Constrained optimization in Chebfun". http://www.chebfun.org/examples/opt/ConstrainedOptimization.html. 
  16. ↑ Simionescu, P.A. (2020). "A collection of bivariate nonlinear optimisation test problems with graphical representations". International Journal of Mathematical Modelling and Numerical Optimisation 10 (4): 365–398. doi:10.1504/IJMMNO.2020.110704. 
  17. ↑ Simionescu, P.A. (2014). Computer Aided Graphing and Simulation Tools for AutoCAD Users (1st ed.). Boca Raton, FL: CRC Press. ISBN 978-1-4822-5290-3. 
  18. ↑ Chankong, Vira; Haimes, Yacov Y. (1983). Multiobjective decision making. Theory and methodology.. North Holland. ISBN 0-444-00710-5. 
  19. ↑ Fonseca, C. M.; Fleming, P. J. (1995). "An Overview of Evolutionary Algorithms in Multiobjective Optimization". Evol Comput 3 (1): 1–16. doi:10.1162/evco.1995.3.1.1. 
  20. ↑ F. Kursawe, “A variant of evolution strategies for vector optimization,” in PPSN I, Vol 496 Lect Notes in Comput Sc. Springer-Verlag, 1991, pp. 193–197.
  21. ↑ Schaffer, J. David (1984). "Multiple Objective Optimization with Vector Evaluated Genetic Algorithms". Proceedings of the First International Conference on Genetic Algorithms. OCLC 20004572. 
  22. ↑ 22.0 22.1 22.2 22.3 22.4 Deb, Kalyan; Thiele, L.; Laumanns, Marco; Zitzler, Eckart (2002). "Scalable multi-objective optimization test problems". Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600). 1. pp. 825–830. doi:10.1109/CEC.2002.1007032. ISBN 0-7803-7282-4. 
  23. ↑ Osyczka, A.; Kundu, S. (1 October 1995). "A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm". Structural Optimization 10 (2): 94–99. doi:10.1007/BF01743536. ISSN 1615-1488. 
  24. ↑ Jimenez, F.; Gomez-Skarmeta, A. F.; Sanchez, G.; Deb, K. (May 2002). "An evolutionary algorithm for constrained multi-objective optimization". Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600). 2. pp. 1133–1138. doi:10.1109/CEC.2002.1004402. ISBN 0-7803-7282-4. 



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