Table 2.

Multimodal benchmark functions.

Function	Dimensions	Range	Optimum value
$F8\left(x\right)={\displaystyle {\sum }_{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)}$	15	[−500, 500]	−2094.9145
F9(x)=∑i=1n[xi2 − 10cos(2πxi)+10]	15	[−5.12, 5.12]	0
$F10\left(x\right)=-20\exp \left(-0.2\sqrt{\left(1/n\right){\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)-\text{exp}\left(\left(1/n\right){\displaystyle {\sum }_{i=1}^n\cos \left(2\pi x_i\right)}\right)+20+e$	15	[−32, 32]	0
$F11\left(x\right)=\left(1/4000\right){\displaystyle {\sum }_{i=1}^n{x}_i^2}-{\displaystyle {\prod }_{i=1}^n\cos \left(x_i/\sqrt{i}\right)}+1$	15	[−600, 600]	0
F12(x)=(π/n){10sin(πyi)+ ∑i=1n−1(yi − 1)2 [1+10sin2(πyi+1)]+(yn − 1)2}+∑i=1nu(xi, 10,100,4)
$y_i=1+\left(\left(x_i+1\right)/4\right)u\left(x_i,a,k,m\right)=\left\{\begin{array}{c}k{\left(x_i-a\right)}^m{x}_i>a\\ 0-a<x_i<a\\ k{\left(-x_i-a\right)}^m{x}_i<-a\end{array}\right.$	15	[−50, 50]	0
F13(x)=0.1{sin2(3πx1)+∑i=1n(xi − 1)2[1+ sin2(3πxi+1)]+ (xn − 1)2[1+sin2(2πxn)]}+∑i=1nu(xi, 5,100,4)	15	[−50, 50]	0