Table 3 Multimodal benchmark functions used in the algorithm performance test experiments.
From: Enhanced artificial hummingbird algorithm with chaotic traversal flight
Name | Formulations | Search ranges | D | values |
|---|---|---|---|---|
Quartic | \(\begin{array}{l}{F_7}(x) = \sum \limits _{i = 1}^n {i \cdot x_i^2 + random[0,1)} \end{array}\) | \([-1.28,1.28]\) | 30 | 0 |
Schwefel | \(\begin{array}{l}{F_8}(x) = - \sum \limits _{i = 1}^n {({x_i}\sin (\sqrt{|{x_i}|} ))} \end{array}\) | \([-500,500]\) | 30 | \({-418.9829 \times D }\) |
Rastrigin | \(\begin{array}{l}{F_9}(x) = - \sum \limits _{i = 1}^n {[x_i^2 - 10\cos (2\pi {x_i}) + 10]} \end{array}\) | \([-5.12,5.12]\) | 30 | 0 |
Ackley | \(\begin{array}{l}{F_{10}}(x) = - 20\exp ( - 0.2\sqrt{\frac{1}{n}\sum \limits _{i = 1}^n {x_i^2} } ) \\ - \exp (\frac{1}{n}\sum \limits _{i = 1}^n {\cos 2\pi {x_i}} ) + 20 + e\end{array}\) | \([-32,32]\) | 30 | 0 |
Griewank | \(\begin{array}{l}{F_{11}}(x) = \frac{1}{{400}}\sum \nolimits _{i = 1}^n {x_i^2} - \prod \nolimits _{i = 1}^n {\cos (\frac{{{x_i}}}{{\sqrt{i} }})} + 1\end{array}\) | [-600,600] | 30 | 0 |
Penalized | \(\begin{array}{l}{F_{12}}(x) = \frac{\pi }{n} \cdot 10{\sin ^2}(3\pi {y_1}) + \frac{\pi }{n} \cdot \sum \limits _{i = 1}^{n - 1} {{({y_i} - 1)}^2} \cdot [1 + 10{{\sin }^2}(\pi {y_i} + 1)] + \frac{\pi }{n}\\ \cdot {{({y_n} - 1)}^2} + \sum \limits _{i = 1}^{30} {u({x_i},10,100,4)} \end{array}\) | \([-50,50]\) | 30 | 0 |
Penalized2 | \(\begin{array}{l}{F_{13}}(x) = 0.1 {{\sin }^2}(3\pi {x_1}) + 0.1\sum \limits _{i = 1}^{29} {{({x_i} - 1)}^2} \\\cdot p \cdot [1 + {{\sin }^2}(3\pi {x_{i + 1}})] + 0.1{{({x_n} - 1)}^2}[1 + {{\sin }^2}(2\pi {x_{30}})] + \sum \limits _{i = 1}^{30} {u({x_i},5,100,4)}\end{array}\) | \([-50,50]\) | 30 | 0 |
