Table 5 Fixed-dimension multimodal benchmark functions
Function | Dim | Range | f min |
|---|---|---|---|
\( {f}_{14}(x)={\left(\frac{1}{500}+{\sum}_{j=1}^{25}\frac{1}{j+{\sum}_{i=1}^2{\left({x}_i-{a}_{ij}\right)}^6}\right)}^{-1} \) | 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] | 0.00030 |
\( {f}_{16}(x)=4{x}_1^2-2.1{x}_1^4+\frac{1}{3}{x}_1^6+{x}_1{x}_2-4{x}_2^2+4{x}_2^4 \) | 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 |
\( {\displaystyle \begin{array}{l}{f}_{18}(x)=\left[1+{\left({x}_1+{x}_2+1\right)}^2\left(19-14{x}_1+3{x}_1^2-14{x}_2+6{x}_1{x}_2+3{x}_2^2\right)\right]\\ {}\times \left[30+{\left(2{x}_1-3{x}_2\right)}^2\times \left(18-32{x}_1+12{x}_1^2+48{x}_2-36{x}_1{x}_2+27{x}_2^2\right)\right]\end{array}} \) | 2 | [− 2,2] | 3 |
\( {f}_{19}(x)=-{\sum}_{i=1}^4{c}_i\exp \left(-{\sum}_{j=1}^3{a}_{ij}{\left({x}_j-{p}_{ij}\right)}^2\right) \) | 3 | [1, 3] | −3.86 |
\( {f}_{20}(x)=-{\sum}_{i=1}^4{c}_i\exp \left(-{\sum}_{j=1}^6{a}_{ij}{\left({x}_j-{p}_{ij}\right)}^2\right) \) | 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 |