Table 3.

Multimodal benchmark functions.

Function	Expression	n	Range	fmin
Schwefel	$f_8(x)={\displaystyle {\sum }_{i=1}^n-x_i\sin (\sqrt{\left|x_i\right|})}$	5	[−500,500]	−418.9829 × 5
Rastrigrin	$f_9(x)={\displaystyle {\sum }_{i=1}^n[x_i{}^2-10\cos (2\pi x_i)+10]}$	5	[−5.12,5.12]	0
Ackley	$\begin{array}{l}{f}_{10}(x)=-20\exp (-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i}{}^2})\\ -\exp (\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\cos (2\pi x_i)})+20+e\end{array}$	5	[−32.32]	0
Griewank	$\begin{array}{l}{f}_{11}(x)=\frac{1}{4000}{\displaystyle {\sum }_{i=1}^n{x}_i}{}^2\\ -{\displaystyle {\prod }_{i=1}^n\cos (\frac{x_i}{\sqrt{i}})}+1\end{array}$	5	[−600,600]	0
Penalty#	$\begin{array}{l}{f}_{12}(x)=\frac{\pi }{n}\{10\sin (\pi y_1)+{\displaystyle {\sum }_{i=1}^{n-1}(y_i-1}{)}^2[1\\ +\sin (\pi y_i+1)]+{(y_n-1)}^2\}\\ +{\displaystyle {\sum }_{i=1}^{n}u(x_i,10,100,4)}\}\\ yi=1+\frac{x_i+1}{4}\\ u(x_i,a,k,m)=\{\begin{array}{c}k{(x_i-a)}^m,x_i>a\\ 0,-a<x_i<a\\ k{(-x_i-a)}^m,x_i<-a\end{array}\end{array}$	5	[−50,50]	0
Penalized 1.2	$\begin{array}{l}{f}_{13}(x)=0.1\{{\sin }^2(3\pi x_1)+{\displaystyle {\sum }_{i=1}^n(x_i}-1{)}^2[1\\ +{\sin }^2(3\pi x_i+1)]+{(x_n-1)}^2[1+{\sin }^2(2\pi x_n)]\}\\ +{\displaystyle {\sum }_{i=1}^{n}u(x_i,5,100,4)}\end{array}$	5	[−50,50]	0