Table 3 Benchmark functions \(F_{{{14}}} (x) - F_{{{23}}} (x)\) with fixed dimension.

\(F_{{1{4}}} (x) = \left( {\frac{{1}}{{{500}}} + \sum\limits_{j = 1}^{25} {\frac{1}{{j + \sum\limits_{i = 1}^{2} {\left( {x_{i} - a_{ij} } \right)^{6} } }}} } \right)^{ - 1}\)

2

[− 65.536, 65.536]

0.998

\(F_{{{15}}} (x) = \sum\limits_{i = 1}^{{{11}}} {\left[ {a_{i} - \frac{{x_{1} (b_{i}^{2} + b_{i} x_{i} )}}{{b_{i}^{2} + b_{i} x_{3} + x_{4} }}} \right]^{2} }\)

4

[− 5, 5]

0.0030

\(F_{{{16}}} (x) = {4}x_{1}^{2} - 2.1x_{1}^{4} + \frac{1}{3}x_{1}^{6} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4}\)

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

\([ - 5,10] \times [10,15]\)

0.398

\(\begin{aligned} F_{{{18}}} (x) & = \left[ {{1} + \left( {x_{1} + x_{2} + 1} \right)^{2} \left( {19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} + 6x_{1} x_{2} + 3x_{2}^{2} } \right)} \right] \\ & \quad \times \left[ {30 + \left( {2x_{1} - 3x_{2} } \right)^{2} \left( {18 - 32x_{1} + 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} } \right)} \right] \\ \end{aligned}\)

2

[− 2, 2]

3

\(F_{{{19}}} (x) = - \sum\limits_{i = 1}^{4} {c_{i} \exp \left( { - \sum\limits_{j = 1}^{3} {a_{ij} (x_{j} - p_{ij} )^{2} } } \right)}\)

3

[0, 1]

− 3.86

\(F_{{{20}}} (x) = - \sum\limits_{i = 1}^{4} {c_{i} \exp \left( { - \sum\limits_{j = 1}^{6} {a_{ij} (x_{j} - p_{ij} )^{2} } } \right)}\)

6

[0, 1]

− 3.32

\(F_{21} (x) = - \sum\limits_{i = 1}^{5} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }\)

4

[0, 10]

− 10.1532

\(F_{{2{2}}} (x) = - \sum\limits_{i = 1}^{{7}} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }\)

4

[0, 10]

− 10.4028

\(F_{{2{3}}} (x) = - \sum\limits_{i = 1}^{{{10}}} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }\)

4

[0, 10]

− 10.5363