Table 2.

Considered benchmark functions for the performance analysis.

Cat.	Func.	Description	Range
Unimodal	F1	$F\left(x\right)={\displaystyle \sum _{i=1}^n}{x}_i^2$	[−100, 100]
	F2	$F\left(x\right)={\displaystyle \sum _{i=0}^n}\left|x_i\right|+\left[r\right]{\displaystyle \prod _{i=0}^n}\left|x_i\right|$	[−10, 10]
	F3	$F\left(x\right)={\displaystyle \sum _{i=1}^d}{\left({\displaystyle \sum _{j=1}^i}{x}_j\right)}^2$	[−100, 100]
	F4	$F\left(x\right)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left(x_i^2-x_{i+1}\right)}^2+{\left(1-x_i\right)}^2\right]$	[−30, 30]
	F5	$F\left(x\right)={\displaystyle \sum _{i=1}^n}{\left(\left[x_i+0.5\right]\right)}^2$	[−100, 100]
	F6	$F\left(x\right)={\displaystyle \sum _{i=0}^n}i{x}_i^4+random(0,1)$	[−128, 128]
Multimodal	F7	$F\left(x\right)={\displaystyle \sum _{i=1}^n}\left(-x_{i}sin\left(\sqrt{\left|x_i\right|}\right)\right)$	[−500, 500]
	F8	$\begin{array}{c}\hfill F\left(x\right)=-20exp\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n}{x}_i^2}\right)-exp\left(\frac{1}{n}{\displaystyle \sum _{i=1}^n}cos\left(2\pi x_i\right)\right)+20+e\end{array}$	[−32, 32]
	F9	$\begin{array}{cc}\hfill F\left(x\right)=& \frac{\pi }{n}\left\{10sin\left(\pi y_1\right)\right\}+{\displaystyle \sum _{i=1}^{n-1}}{\left(y_i-1\right)}^2[1+10{sin}^2\left(\pi y_{i+1}\right)+{\displaystyle \sum _{i=1}^n}u\left(x_i,10,100,4\right),\hfill \\ & \mathrm{where}{y}_i=1+\frac{x_i+1}{4},u\left(x_i,a,k,m\right)\left\{\begin{array}{cc}K{\left(x_i-a\right)}^m\hfill & x_i>a\hfill \\ 0\hfill & -a\le x_i\ge a\hfill \\ K{\left(-x_i-a\right)}^m\hfill & -a\le x_i\hfill \end{array}\right.\hfill \end{array}$	[−50, 50]
	F10	$\begin{array}{cc}\hfill F\left(x\right)=& 0.1\left({sin}^2\left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^n}{\left(x_i-1\right)}^2\left[1+{sin}^2\left(3\pi x_i+1\right)\right]+{\left(x_n-1\right)}^2\left[1+{sin}^2\left(2\pi x_n\right)\right]\right)\hfill \\ & +{\displaystyle \sum _{i=1}^n}u\left(x_i,5,100,4\right)\hfill \end{array}$	[−50, 50]
	F11	$F\left(x\right)={\left(\frac{1}{500}+{\displaystyle \sum _{j=1}^{25}}\frac{1}{j+\sum _{i=1}^2}\left(x_i-a_{ij}\right)\right)}^{-1}$	[−65,65]
	F12	$F\left(x\right)={\displaystyle \sum _{i=1}^{11}}{\left[a_i-\frac{x_1\left(b_i^2+b_i{x}_2\right)}{b_i^2+b_i{x}_3+x_4}\right]}^2$	[−5, 5]
	F13	$F\left(x\right)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	[−5, 5]
	F14	$\begin{array}{c}\hfill F\left(x\right)={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos{x}_1+10\end{array}$	[−5, 5]
	F15	$\begin{array}{c}\hfill F\left(x\right)={\displaystyle \sum _{i=1}^{d/4}}\left[{\left(x_{4i-3}+10x_{4i-2}\right)}^2+5{\left(x_{4i-1}-x_{4i}\right)}^2+{\left(x_{4i-2}-2x_{4i-1}\right)}^4+10{\left(x_{4i-3}-x_{4i}\right)}^4\right]\end{array}$	[−4, 5]
	F16	$F\left(x\right)=-{\displaystyle \sum _{i=1}^4}{c}_{i}exp\left(-{\displaystyle \sum _{i=1}^3}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	[−1, 2]
	F17	$F\left(x\right)=-{\displaystyle \sum _{i=1}^4}{c}_{i}exp\left(-{\displaystyle \sum _{i=1}^6}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	[0, 1]
	F18	$F\left(x\right)=-{\displaystyle \sum _{i=1}^5}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	[0, 1]
Hybrid	F19	$\begin{array}{c}\hfill F\left(x\right)=-\left(x_2+47\right)sin\left(\sqrt{\left|x_2+\frac{x_1}{2}+47\right|}\right)\\ \hfill -x_{1}sin\left(\sqrt{\left|x_1-\left(x_2+47\right)\right|}\right)\end{array}$	[−512, 512]
	F20	$F\left(x\right)=-\left|sin\left(x_1\right)cos\left(x_2\right)exp\left(\left|1-\frac{\sqrt{x_1^2+x_2^2}}{\pi }\right|\right)\right|$	[−10, 10]
Hybrid	F21	$\begin{array}{c}\hfill F\left(x\right)=\left({\displaystyle \sum _{i=1}^5}icos\left((i+1)x_1+i\right)\right)\times \left({\displaystyle \sum _{i=1}^5}icos\left((i+1)x_2+i\right)\right)\end{array}$	[−5.12, 5.12]
	F22	$F\left(x\right)=sin\left(x_1+x_2\right)+{\left(x_1-x_2\right)}^2-1.5x_1+2.5x_2+1$	[−1.5, 4]
	F23	$F\left(x\right)=\left(4-2.1x_1^2+\frac{x_1^4}{3}\right)x_1^2+x_1{x}_2+\left(-4+4x_2^2\right)x_2^2$	[−3, 3]
	F24	$F\left(x\right)=-cos\left(x_1\right)cos\left(x_2\right)exp\left(-{\left(x_1-\pi \right)}^2-{\left(x_2-\pi \right)}^2\right)$	[−100, 100]
	F25	$F\left(x\right)=a{\left(x_2-b{x}_1^2+c{x}_1-r\right)}^2+s(1-t)cos\left(x_1\right)+s$	[−5, 10]
	F26	$\begin{array}{cc}\hfill F\left(x\right)=& \left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)\right]\hfill \\ & \times \left[30+{\left(2x_1-3x_2\right)}^2\left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)\right]\hfill \end{array}$	[−2, 2]
	F27	$F\left(x\right)=\frac{1}{2}{\displaystyle \sum _{i=1}^d}\left(x_i^4-16x_i^2+5x_i\right)$	[−5, 5]
	F28	$\begin{array}{cc}\hfill F\left(x\right)=& {sin}^2\left(\pi w_1\right)+{\displaystyle \sum _{i=1}^{d-1}}{\left(w_i-1\right)}^2\left[1+10{sin}^2\left(\pi w_i+1\right)\right]+{\left(w_d-1\right)}^2\left[1+{sin}^2\left(2\pi w_d\right)\right],\hfill \\ & \mathrm{where}{w}_i=1+\frac{x_i-1}{4},\mathrm{for}\mathrm{all}i=1,\dots ,d\hfill \end{array}$	[−10, 10]
	F29	$\begin{array}{cc}\hfill F\left(x\right)=& 100{\left(x_1^2-x_2\right)}^2+{\left(x_1-1\right)}^2+{\left(x_3-1\right)}^2+90{\left(x_3^2-x_4\right)}^2\hfill \\ & +10.1\left({\left(x_2-1\right)}^2+{\left(x_4-1\right)}^2\right)+19.8\left(x_2-1\right)\left(x_4-1\right)\hfill \end{array}$	[−10, 10]
	F30	$\begin{array}{cc}\hfill F\left(x\right)=& -\sum _{i=1}^4{\alpha }_{i}exp\left(-\sum _{j=1}^3{A}_{ij}{\left(x_j-P_{ij}\right)}^2\right),\mathrm{where}\alpha ={(1.0,1.2,3.0,3.2)}^T,\hfill \\ & \mathbf{A}=\left(\begin{array}{ccc}3.0\hfill & 10\hfill & 30\hfill \\ 0.1\hfill & 10\hfill & 35\hfill \\ 3.0\hfill & 10\hfill & 30\hfill \\ 0.1\hfill & 10\hfill & 35\hfill \end{array}\right),\mathbf{P}={10}^{-4}\left(\begin{array}{ccc}3689& 1170& 2673\\ 4699& 4387& 7470\\ 1091& 8732& 5547\\ 381& 5743& 8828\end{array}\right).\hfill \end{array}$	[0, 1]
	F31	$\begin{array}{cc}\hfill F\left(x\right)=& -\sum _{i=1}^m{\left(\sum _{j=1}^4{\left(x_j-C_{ji}\right)}^2+{\beta }_i\right)}^{-1},\mathrm{where}m=10,\beta =\frac{1}{10}{(1,2,2,4,4,6,3,7,5,5)}^T,\hfill \\ & \mathbf{C}=\left(\begin{array}{cccccccccc}4.0\hfill & 1.0\hfill & 8.0\hfill & 6.0\hfill & 3.0\hfill & 2.0\hfill & 5.0\hfill & 8.0\hfill & 6.0\hfill & 7.0\hfill \\ 4.0\hfill & 1.0\hfill & 8.0\hfill & 6.0\hfill & 7.0\hfill & 9.0\hfill & 3.0\hfill & 1.0\hfill & 2.0\hfill & 3.6\hfill \\ 4.0\hfill & 1.0\hfill & 8.0\hfill & 6.0\hfill & 3.0\hfill & 2.0\hfill & 5.0\hfill & 8.0\hfill & 6.0\hfill & 7.0\hfill \\ 4.0\hfill & 1.0\hfill & 8.0\hfill & 6.0\hfill & 7.0\hfill & 9.0\hfill & 3.0\hfill & 1.0\hfill & 2.0\hfill & 3.6\hfill \end{array}\right).\hfill \end{array}$	[0, 10]
	F32	$\begin{array}{cc}\hfill F\left(x\right)=& -\sum _{i=1}^4{\alpha }_{i}exp\left(-\sum _{j=1}^6{A}_{ij}{\left(x_j-P_{ij}\right)}^2\right),\mathrm{where}\alpha ={(1.0,1.2,3.0,3.2)}^T,\hfill \\ & \mathbf{A}=\left(\begin{array}{cccccc}10& 3& 17& 3.50& 1.7& 8\\ 0.05& 10& 17& 0.1& 8& 14\\ 3& 3.5& 1.7& 10& 17& 8\\ 17& 8& 0.05& 10& 0.1& 14\end{array}\right),\hfill \\ \hfill & \mathbf{P}={10}^{-4}\left(\begin{array}{cccccc}1312& 1696& 5569& 124& 8283& 5886\\ 2329& 4135& 8307& 3736& 1004& 9991\\ 2348& 1451& 3522& 2883& 3047& 6650\\ 4047& 8828& 8732& 5743& 1091& 381\end{array}\right).\hfill \end{array}$	[0, 1]
	F33	$F\left(x\right)=-{\displaystyle \sum _{i=1}^d}sin\left(x_i\right){sin}^{2m}\left(\frac{i{x}_i^2}{\pi }\right)$	[0, $\pi$]