Table 3 Description of unimodal and multimodal benchmark functions.
Func | Name | Function’s expressions | Search range | \(f_{min}\) | D | |
|---|---|---|---|---|---|---|
Unimodal functions | F1 | Sphere Problem | \( f_1=\sum _{i=1}^n{ x_i ^2 } \) | \([-100,100]^n\) | 0 | 50 |
F2 | Schwefel’s Problem 2.22 | \( f_2=\sum _{i=1}^n{\left| x_i \right| }+\prod _{i=1}^n{\left| x_i \right| } \) | \([-10,10]^n\) | 0 | 50 | |
F3 | Schwefel’s Problem 1.2 | \( f_3=\sum _{i=1}^n{ \left( \sum _{j=1}^i{ x_j }\right) ^2} \) | \([-100,100]^n\) | 0 | 50 | |
F4 | Schwefel’s Problem 2.21 | \( f_4=\max \left\{ \left| x_i \right| ,1\le i\le n \right\} \) | \([-100,100]^n\) | 0 | 50 | |
F5 | Rosenbrock ’s Problem | \( f_{5}=\sum _{i=1}^{n-1}\biggl [100\biggl (x_{i+1}-x_{i}^{2}\biggr )^{2}+\biggl (x_{i}-1\biggr )^{2}\biggr ]\) | \([-30,30]^n\) | 0 | 50 | |
F6 | Step Problem | \( f_{6}=\sum _{i=1}^{n}{\big (}{\big [}x_{i}+0.5{\big ]}{\big )}^{2} \) | \([-100,100]^n\) | 0 | 50 | |
F7 | Quartic Noise | \(f_3=\sum _{i=1}^n{ix_{i}^{4}}+random\left[ 0.1 \right) \) | \([-1.28,1.28]^n\) | 0 | 50 | |
Multimodal functions | F8 | Rastrigin | \( f_8=\sum _{i=1}^n{\left[ x_{i}^{2}-10\cos \left( 2\pi x_i \right) +10 \right] } \) | \([-5.12,5.12]^n\) | 0 | 50 |
F9 | Ackley | \( f_{9}=20+e-20\exp \left( -0.2\sqrt{\frac{1}{n}\sum _{i= 1}^{n}x_i^2}\right) -\exp \left( \frac{1}{n}\sum _{i=1}^{n}\cos \left( 2\pi x_i\right) \right) \) | \([-32,32]^n\) | 0 | 50 | |
F10 | Griewank | \( f_{10}={\frac{1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}\cos \left( {\frac{x_{i}}{\sqrt{i}}}\right) +1 \) | \([-600,600]^n\) | 0 | 50 | |
F11 | Penalized’s Function | \( f_{11}= \frac{\pi }{n}\bigg \{10{\text {sin}}\big (\pi x_{i}\big )+\sum _{i=1}^{n-}\big (x_{i}-1\big )^{2}\bigg [1+10{\text {sin}}^{2}\big (\pi x_{i+i}\big )\bigg ]+\big (x_{n}-1\big )^{2}\bigg \} +\sum _{i=1}^{n}u\big (x_{i},10,100,4\big ) \) | \([-50,50]\) | 0 | 50 | |
F12 | Penalized’s Function | \( f_{12}\left( x \right) = \frac{1}{10} \sin ^2\left( 3\pi x_1 \right) + \frac{1}{10} \sum _{k=1}^n{\left( x_k-1 \right) }^2\left[ 1+\sin ^2\left( 3\pi x_l+1 \right) \right] + \sum _{i=1}^n{u}\left( x_i,5,100,4 \right) + \frac{1}{10} \left( x_n-1 \right) ^2\left[ 1+\sin ^2\left( 2\pi x_n \right) \right] \) | \([-50,50]\) | 0 | 50 |
