Table 5.

Fixed-dimension multimodal benchmark functions.

Function	Dim	Range	fmin
$f_{14}(x)={\left(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2{\left(x_i-a_{\mathit{ij}}\right)}^6}\right)}^{-1}$	2	[−65,65]	1
$f_{15}(x)={\sum }_{i=1^{11}\left[a_i-\frac{x_1\left(b_i^2+b_i{x}_2\right)}{b_i^2+b_i{x}_3+x_4}\right]}^2$	4	[−5, 5]	0.0003
$f_{16}(x)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[−5,5]	−1.0316
$f_{17}(x)={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)\cos x_1+10$	2	[−5,5]	0.398
$f_{18}(x)=\left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)\right]$	2	[−2,2]	3
$f_{19}(x)=-\sum _{i=1}^4{c}_i\exp \left(-\sum _{j=1}^3{a}_{\mathit{ij}}{\left(x_j-p_{\mathit{ij}}\right)}^2\right)$	3	[1,3]	−3.86
$f_{20}(x)=-\sum _{i=1}^4{c}_i\exp \left(-\sum _{j=1}^6{a}_{\mathit{ij}}{\left(x_j-p_{\mathit{ij}}\right)}^2\right)$	6	[0,1]	−3.32
$f_{21}(x)=-\sum _{i=1}^5{\left[\left(X-a_i\right){\left(X-a_i\right)}^{\mathrm{T}}+c_i\right]}^{-1}$	4	[0,10]	−10.1532
$f_{22}(x)=-\sum _{i=1}^7{\left[\left(X-a_i\right){\left(X-a_i\right)}^{\mathrm{T}}+c_i\right]}^{-1}$	4	[0,10]	−10.4028
$f_{23}(x)=-\sum _{i=1}^{10}{\left[\left(X-a_i\right){\left(X-a_i\right)}^{\mathrm{T}}+c_i\right]}^{-1}$	4	[0,10]	−10.5363