Table 2.

10 benchmark test functions for validation.

Function	Dim	Range	fmin
$f_1\left(x\right)={\sum }_{i=1}^n{x}_i^2$	30	[− 100, 100]	0
$f_2\left(x\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	30	[− 10, 10]	0
$f_3\left(x\right)={\sum }_{i=1}^n{\left({\sum }_{j-1}^i,x_j\right)}^2$	30	[− 100, 100]	0
$f_4\left(x\right)={\text{max}}_i\left\{\left|x_i\right|,\text{,},1,\le ,i,\le ,n\right\}$	30	[− 100, 100]	0
$f_5(x)={\sum }_{i=1}^n[x_i^2-10\text{cos}(2\pi x_i)+10]$	30	[− 5.12, 5.12]	0
$f_6(x)=-20\text{exp}\left(-,0.2,\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2}\right)-\text{exp}\left(\frac{1}{n},{\sum }_{i=1}^n,\text{cos},(2\pi x_i)\right)+20+e$	30	[− 32, 32]	0
$f_7(x)=\frac{1}{4000}{\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^n\text{cos}\left(\frac{x_i}{\sqrt{i}}\right)+1$	30	[− 600, 600]	0
$f_8(x)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[− 5, 5]	− 1.0316
$f_9(x)={\left(x_2,-,\frac{5.1}{4{\pi }^2},x_1^2,+,\frac{5}{\pi },x_1,-,6\right)}^2+10\left(1,-,\frac{1}{8\pi }\right)\text{cos}{x}_1+10$	2	[− 5, 5]	0.398
$f_{10}(x)=-{\sum }_{i=1}^{10}{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	4	[0, 10]	− 10.5363