Table 2. The Benchmark functions.

Function ID	Function Name	Equation	Function Typ	Dimension	f(x*)	Bounds of X
F1	Sphere	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^n{x}_i^2}$	Unimodal	30	0	UB(100)
						LB(-100)
F2	Schwefel’s problem 2.22	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^n\left|x_i\right|}+{\displaystyle {\prod }_{i=1}^n\left|x_i\right|}$	Unimodal	30	0	UB(10)
						LB(-10)
F3	Quartic	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^{n}i{x}_i^4}+\mathrm{rand}()$	Unimodal	30	0	UB(1.28)
						LB(-1.28)
F4	Sum squares	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^{n}i{x}_i^2}$	Unimodal	30	0	UB(10)
						LB(-10)
F5	Sum of different power	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^n{\left|x_i\right|}^{i+1}}$	Unimodal	30	0	UB(1)
						LB(-1)
F6	Rosenbrock	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^{n-1}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right)}$	Unimodal	30	0	UB(30)
						LB(30)
F7	Ackley	$\begin{array}{l}\mathrm{f}(x)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)-\\ \exp \left(\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\cos \left(2\pi x_i\right)}\right)+20+e\end{array}$	Multimodal	30	0	UB(32)
						LB(-32)
F8	Griewank	$\mathrm{f}(x)=\frac{1}{4000}{\displaystyle {\sum }_{i=1}^{n-1}{x}_i^2-{\displaystyle {\prod }_{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)}}+1$	Multimodal	30	0	UB(600)
						LB(600)
F9	Alpine	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^n\left\{x_i\sin \left(x_i\right)+0.1x_i\right\}}$	Multimodal	30	0	UB(10)
						LB(10)
F10	Powell	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^{n/4}\left\{\begin{array}{l}{\left(x_{4i-3}+10x_{4i-2}\right)}^2+4{\left(x_{4i-1}+x_{4i}\right)}^2+\\ {\left(x_{4i-2}+2x_{4i-1}\right)}^2+10{\left(x_{4i-3}+x_{4i}\right)}^2\end{array}\right\}}$	Multimodal	30	0	UB(5)
						LB(-4)
F11	Rastrigin	$\mathrm{f}(x)={\displaystyle {\sum }_{i=1}^n\left(x_i^2-10\cos \left(2\pi x_i\right)+10\right)}$	Multimodal	30	0	UB = 5.12
						LB(-5.12)
F12	Solomon problem	$\mathrm{f}(x)=1-\cos \left(2\pi \sqrt{{\displaystyle {\sum }_{i=1}^n{x}_i^2}}\right)+0.1\sqrt{{\displaystyle {\sum }_{i=1}^n{x}_i^2}}$	Multimodal	30	0	UB(100)
						LB(-100)