Table 3 Description of fixed-dimension multimodal benchmark functions.

Shekel’s Foxholes function

\(F_{10}(x) = \sum _{j=1}^{25} \frac{1}{j + \sum _{i=1}^{2} (x_i - a_{ij})^6}\)

2

[− 65, 65]

1

Kowalik function

\(F_{11}(x) = \sum _{i=1}^{11} \left( y_i - \frac{x_1(u_i^2 + u_i x_2)}{u_i^2 + u_i x_3 + x_4}\right) ^2\)

4

[− 5, 5]

0.000307

Six-Hump Camelback function

\(F_{12}(x) = (4 - 2.1x_1^2 + \frac{x_1^4}{3})x_1^2 + x_1 x_2 + (-4 + 4x_2^2)x_2^2\)

2

[− 5, 5]

− 1.0316

Branin RCOS function

\(F_{13}(x) = a(x_2 - b x_1^2 + cx_1 - r)^2 + s(1-t)\cos (x_1) + s\)

2

[− 5, 5]

0.398

Goldstein-Price function

\(\begin{aligned} F_{14}(x) =& \left( 1 + (x_1 + x_2 + 1)^2(19 - 14x_1 + 3x_1^2 - 14x_2 + 6x_1x_2 + 3x_2^2)\right) \\ & \times \left( 30 + (2x_1 - 3x_2)^2(18 - 32x_1 + 12x_1^2 + 48x_2 - 36x_1x_2 + 27x_2^2)\right) \end{aligned}\)

2

[− 2.2, 2.2]

3