Table 2.

Benchmark functions.

Function	Dim	Scope	Solution
$f_1={\displaystyle \sum _{i=1}^n{x}_i^2}$	30	[−100, 100]	0
$f_2={\displaystyle \sum _{i=1}^n{x}_i+{\displaystyle \prod _{i=1}^n\left|x_i\right|}}$	30	[−10, 10]	0
$f_3={{\displaystyle \sum _{i=1}^n\left({\displaystyle \sum _{j-1}^i{x}_j}\right)}}^2$	30	[−100, 100]	0
$f_4={\max }_i\left\{\right|x_i|,1\le i\le n\}$	30	[−100, 100]	0
$f_5={\displaystyle {\sum }_{i=1}^{n-1}[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2]}$	30	[−30, 30]	0
$f_6={\displaystyle {\sum }_{i=1}^d{(⌊x_i+0.5⌋)}^2}$	30	[−100, 100]	0
$f_7={\displaystyle \sum _{i=1}^{n}i{x}_i^4+random[0,1)}$	30	[−1.28, 1.28]	0
$f_8={\displaystyle \sum _{i=1}^n[x_i^2-10\cos (2\pi x_i^{})+10]}$	30	[−5.12, 5.12]	0
$\begin{array}{l}{f}_9=-20\exp (-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i^2}})\\ -\exp (\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\cos (2\pi x_i^{})}+20+e)\end{array}$	30	[−32, 32]	0
$f_{10}=\frac{1}{4000}{\displaystyle \sum _{i=1}^n{x}_i^2-{\displaystyle {\prod }_{i=1}^d\cos (\frac{x_i}{\sqrt{i}})}+1]}$	30	[−600, 600]	0
$\begin{array}{l}{f}_{11}=\frac{\pi }{n}\{10\sin (\pi y_1)+{\displaystyle \sum _{i=1}^{D-1}{(y_i-1)}^2[1+10{\sin }^2(\pi y_{i+1})]+{(y_n-1)}^2}\}\\ +{\displaystyle \sum _{i=1}^{D}u(x_i,5,100,4)}\end{array}$	30	[−50, 50]	0.398
$\begin{array}{l}{f}_{12}=0.1\{10\sin (3\pi x_1)+{\displaystyle \sum _{i=1}^{D-1}{(x_i-1)}^2[1+10{\sin }^2(3\pi x_{i+1})]+{(x_n-1)}^2}\}\\ +{\displaystyle \sum _{i=1}^{D}u(x_i,5,100,4)}\end{array}$	30	[−50, 50]	3
$f_{13}={\displaystyle \sum _{i=1}^D|x_i^{}\sin (x_i^{})+0.1x_i^{}|}$	30	[−10, 10]	0
$f_{14}=0.5+({(\sin ({\displaystyle \sum _{i=1}^D{x}_i^2}))}^2-0.5)\cdot {(1+0.001({\displaystyle \sum _{i=1}^D{x}_i^2}))}^{-2}$	30	[−100, 100]	0
$f_{15}=({\displaystyle \sum _{i=1}^D[x_i^2+2x_{i+1}^2-0.3\cos (3\pi x_i^{})-0.4\cos (4\pi x_{i+1}^{})+0.7]}$	30	[−15, 15]	0