Table 6.

Fixed-dimension multimodal benchmark functions.

Function	Dim	Range	f min
$f_{\mathrm{14}}\left(x\right)={\left(\frac{\mathrm{1}}{\mathrm{500}}+\sum _{j=\mathrm{1}}^{\mathrm{25}}\frac{\mathrm{1}}{j+{\sum }_{i=\mathrm{1}}^{\mathrm{2}}{\left(x_i-a_{ij}\right)}^{\mathrm{6}}}\right)}^{-\mathrm{1}}$	2	[-65,65]	1
$f_{\mathrm{15}}\left(x\right)={\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{11}}{\left[a_i-\frac{x_{\mathrm{1}}\left(b_i^{\mathrm{2}}+b_i{x}_{\mathrm{2}}\right)}{b_i^{\mathrm{2}}+b_i{x}_{\mathrm{3}}+x_{\mathrm{4}}}\right]}^2$	4	[-5, 5]	0.00030
$f_{\mathrm{16}}\left(x\right)=\mathrm{4}{x}_{\mathrm{1}}^{\mathrm{2}}-\mathrm{2.1}{x}_{\mathrm{1}}^{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{3}}{x}_{\mathrm{1}}^{\mathrm{6}}+x_{\mathrm{1}}{x}_{\mathrm{2}}-\mathrm{4}{x}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{4}{x}_{\mathrm{2}}^{\mathrm{4}}$	2	[-5,5]	-1.0316
$f_{\mathrm{17}}\left(x\right)={\left(x_{\mathrm{2}}-\frac{\mathrm{5.1}}{\mathrm{4}{\pi }^{\mathrm{2}}}{x}_{\mathrm{1}}^{\mathrm{2}}+\frac{\mathrm{5}}{\pi }{x}_{\mathrm{1}}-\mathrm{6}\right)}^{\mathrm{2}}+\mathrm{10}\left(\mathrm{1}-\frac{\mathrm{1}}{\mathrm{8}\pi }\right)\cos x_{\mathrm{1}}+\mathrm{10}$	2	[-5,5]	0.398
f 18(x) = [1 + (x1 + x2 + 1)2(19 − 14x1 + 3x12 − 14x2 + 6x1x2 + 3x22)]	2	[-2,2]	3
×[30 + (2x1 − 3x2)2times(18 − 32x1 + 12x12 + 48x2 − 36x1x2 + 27x22)]
$f_{\mathrm{19}}\left(x\right)=-{\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{4}}{c}_i\exp \left(-{\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{3}}{a}_{ij}{\left(x_j-p_{ij}\right)}^{\mathrm{2}}\right)$	3	[1,3]	-3.86
$f_{\mathrm{20}}\left(x\right)=-{\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{4}}{c}_i\exp \left(-{\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{6}}{a}_{ij}{\left(x_j-p_{ij}\right)}^{\mathrm{2}}\right)$	6	[0,1]	-3.32
$f_{\mathrm{21}}\left(x\right)=-{\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{5}}{\left[\left(X-a_i\right){\left(X-a_i\right)}^{\mathrm{T}}+c_i\right]}^{-1}$	4	[0,10]	-10.1532
$f_{\mathrm{22}}\left(x\right)=-{\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{7}}{\left[\left(X-a_i\right){\left(X-a_i\right)}^{\mathrm{T}}+c_i\right]}^{-\mathrm{1}}$	4	[0,10]	-10.4028
$f_{\mathrm{23}}\left(x\right)=-{\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{10}}{\left[\left(X-a_i\right){\left(X-a_i\right)}^{\mathrm{T}}+c_i\right]}^{-\mathrm{1}}$	4	[0,10]	-10.5363