Problems.md

Test Functions

Bellow You'll find the definitions of all the test functions implemented in this package.

Ackley

Function name: ackley

$$f(x)=-20e^{-0.2\sqrt{D^{-1}{\sum }_{i=1}^D{x}_i^2}}-e^{D^{-1}{\sum }_{i=1}^D\cos (2\pi x_i)}+20+e$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Alpine 1

Function name: alpine1

$$f(x)=\sum _{i=1}^D|x_i\sin \left(x_i\right)+0.1x_i|$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Alpine 2

Function name: alpine2

$$f(x)=\prod _{i=1}^D\sqrt{x_i}\sin (x_i)$$

Dimensions: $D$

Global optimum: $f(x^{\ast })={2.808}^D$ for $x_i^{\ast }=7.917$

Cigar

Function name: cigar

$$f(x)=x_1^2+{10}^6\sum _{i=2}^D{x}_i^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Cosine Mixture

Function name: cosine_mixture

$$f(x)=-0.1\sum _{i=1}^D\cos (5\pi x_i)-\sum _{i=1}^D{x}_i^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=-0.1D$ for $x_i^{\ast }=0$

Csendes

Function name: csendes

$$f(x)=\sum _{i=1}^D{x}_i^6(2+\sin \frac{1}{x_i})$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Dixon-Price

Function name: dixon_price

$$f(x)=(x_1-1{)}^2+\sum _{i=2}^{D}i(2x_i^2-x_{i-1}{)}^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=2^{-\frac{(2^i-2)}{2^i}}$

Griewank

Function name: griewank

$$f(x)=\sum _{i=1}^D\frac{x_i^2}{4000}-\prod _{i=1}^D\cos (\frac{x_i}{\sqrt{i}})+1$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Katsuura

Function name: katsuura

$$\prod _{i=1}^D(1+i\sum _{j=1}^{32}\frac{|2^j{x}_i-round\left(2^j{x}_i\right)|}{2^j})$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=1$ for $x_i^{\ast }=0$

Levy

Function name: levy

$$\begin{array}{c}{\sin }^2(\pi w_1)+\sum _{i=1}^{D-1}(w_i-1{)}^2(1+10{\sin }^2(\pi w_i+1))+(w_d-1{)}^2(1+{\sin }^2(2\pi w_d)),\text{where}\\ w_i=1+\frac{x_i-1}{4},\text{for all }i=1,\dots ,D\end{array}$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=1$

Michalewicz

Function name: michalewicz

$$f(x)=-\sum _{i=1}^D\sin (x_i){\sin }^{2m}\left(\frac{i{x}_i^2}{\pi }\right)$$

Dimensions: $D$

Global optimum: $\text{at }D=2,f(x^{\ast })=-1.8013$ for $x^{\ast }=(2.20,1.57)$

Perm 1

Function name: perm1

$$f(x)=\sum _{i=1}^D{(\sum _{j=1}^D(j^i+\beta )({\left(\frac{x_j}{j}\right)}^i-1))}^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=i$

Perm 2

Function name: perm2

$$f(x)=\sum _{i=1}^D{(\sum _{j=1}^D(j-\beta )(x_j^i-\frac{1}{j^i}))}^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=\frac{1}{i}$

Pinter

Function name: pinter

$$f(x)=\sum _{i=1}^{D}i{x}_i^2+\sum _{i=1}^{D}20i{\sin }^{2}A+\sum _{i=1}^{D}i{\log }_{10}(1+i{B}^2),\text{where}$$ $$\begin{array}{rl}A& =(x_{i-1}\sin (x_i)+\sin (x_{i+1}))\\ B& =(x_{i-1}^2-2x_i+3x_{i+1}-\cos (x_i)+1)\end{array}$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Powell

Function name: powell

$$f(x)=\sum _{i=1}^{D/4}[(x_{4i-3}+10x_{4i-2}{)}^2+5(x_{4i-1}-x_{4i}{)}^2+(x_{4i-2}-2x_{4i-1}{)}^4+10(x_{4i-3}-x_{4i}{)}^4]$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Qing

Function name: qing

$$f(x)=\sum _{i=1}^D{(x_i^2-i)}^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=\pm \sqrt{i}$

Quintic

Function name: quintic

$$f(x)=\sum _{i=1}^D|x_i^5-3x_i^4+4x_i^3+2x_i^2-10x_i-4|$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=-1\text{or}{x}_i^{\ast }=2$

Rastrigin

Function name: rastrigin

$$f(x)=10D+\sum _{i=1}^D[x_i^2-10\cos (2\pi x_i)]$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Rosenbrock

Function name: rosenbrock

$$f(x)=\sum _{i=1}^{D-1}[100(x_{i+1}-x_i^2{)}^2+(x_i-1{)}^2]$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=1$

Salomon

Function name: salomon

$$f(x)=1-\cos \left(2\pi \sqrt{{\sum }_{i=1}^D{x}_i^2}\right)+0.1\sqrt{{\sum }_{i=1}^D{x}_i^2}$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Schaffer 2

Function name: schaffer2

$$f(x)=0.5+\frac{{\sin }^2(x_1^2-x_2^2)-0.5}{{[1+0.001(x_1^2+x_2^2)]}^2}$$

Dimensions: 2

Global optimum: $f(x^{\ast })=0$ for $x^{\ast }=(0,0)$

Schaffer 4

Function name: schaffer4

$$f(x)=0.5+\frac{{\cos }^2(\sin (|x_1^2-x_2^2|))-0.5}{{[1+0.001(x_1^2+x_2^2)]}^2}$$

Dimensions: 2

Global optimum: $f(x^{\ast })=0.292579$ for $x^{\ast }=(0,\pm 1.25313)\text{or}(\pm 1.25313,0)$

Schwefel

Function name: schwefel

$$f(x)=418.9829D-\sum _{i=1}^D{x}_i\sin (\sqrt{|x_i|})$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=420.9687$

Schwefel 2.21

Function name: schwefel221

$$f(x)=\underset{1\le i\le D}{max}|x_i|$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Schwefel 2.22

Function name: schwefel222

$$f(x)=\sum _{i=1}^D|x_i|+\prod _{i=1}^D|x_i|$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Sphere

Function name: sphere

$$f(x)=\sum _{i=1}^D{x}_i^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Step

Function name: step

$$f(x)=\sum _{i=1}^D(\lfloor |x_i|\rfloor )$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Step 2

Function name: step2

$$f(x)=\sum _{i=1}^D{(\lfloor x_i+0.5\rfloor )}^2$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=-0.5$

Styblinski-Tang

Function name: styblinski_tang

$$$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=-39.16599D$ for $x_i^{\ast }=-2.903534$

Trid

Function name: trid

$$f(x)=\sum _{i=1}^D{(x_i-1)}^2-\sum _{i=2}^D{x}_i{x}_{i-1}$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=\frac{-D(D+4)(D-1)}{6}$ for $x_i^{\ast }=i(d+1-i)$

Weierstrass

Function name: weierstrass

$$f(x)=\sum _{i=1}^D[\sum _{k=0}^{k_{max}}{a}^k\cos (2\pi b^k(x_i+0.5))]-D\sum _{k=0}^{k_{max}}{a}^k\cos \left(\pi b^k\right)$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

Whitley

Function name: whitley

$$f(x)=\sum _{i=1}^D\sum _{j=1}^D[\frac{(100(x_i^2-x_j{)}^2+(1-x_j{)}^2{)}^2}{4000}-\cos (100(x_i^2-x_j{)}^2+(1-x_j{)}^2)+1]$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=1$

Zakharov

Function name: zakharov

$$f(x)=\sum _{i=1}^D{x}_i^2+{\left(\sum _{i=1}^{D}0.5i{x}_i\right)}^2+{\left(\sum _{i=1}^{D}0.5i{x}_i\right)}^4$$

Dimensions: $D$

Global optimum: $f(x^{\ast })=0$ for $x_i^{\ast }=0$

References

[1] P. Ernesto and U. Diliman, “MVF–Multivariate Test Functions Library in C for Unconstrained Global Optimization,” University of the Philippines Diliman, Quezon City, 2005.

[2] M. Jamil and X.-S. Yang, “A literature survey of benchmark functions for global optimisation problems,” International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, p. 150, Jan. 2013, doi: 10.1504/ijmmno.2013.055204.

[3] J. J. Liang, B. Y. Qu, and P. N. Suganthan, “Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization,” Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, vol. 635, no. 2, 2013.

[4] S. Surjanovic and D. Bingham, Virtual Library of Simulation Experiments: Test Functions and Datasets. Retrieved November 7, 2023, from https://www.sfu.ca/~ssurjano/.