Table 1 Descriptions of the 26 benchmark functions.
Function name | s/n | Formula | Range | Category | \(f_{min}^*\) |
|---|---|---|---|---|---|
Sphere | F1 | \(f_1(x) = \sum _{i=1}^{\text {dim}} x_i^2\) | \([-100, 100]\) | Unimodal | 0 |
Schwefel 2.22 | F2 | \(f_2(x) = \sum _{i=1}^{\text {dim}} |x_i| + \prod _{i=1}^{\text {dim}} |x_i|\) | \([-10, 10]\) | Unimodal | 0 |
Schwefel 1.2 | F3 | \(f_3(x) = \sum _{i=1}^{\text {dim}} \left( \sum _{j=1}^i x_j \right) ^2\) | \([-100, 100]\) | Unimodal | 0 |
Schwefel 2.21 | F4 | \(f_4(x) = \max _i \{|x_i|\}, 1 \le i \le \text {dim}\) | \([-100, 100]\) | Unimodal | 0 |
Step | F5 | \(f_5(x) = \sum _{i=1}^{\text {dim}} \left( x_i + 0.5 \right) ^2\) | \([-100, 100]\) | Unimodal | 0 |
Quartic | F6 | \(f_6(x) = \sum _{i=1}^{\text {dim}} ix_i^4 + \text {rand}\) | \([-1.28, 1.28]\) | Unimodal | 0 |
Exponential | F7 | \(f_7(x) = \sum _{i=1}^{\text {dim}} (e^{x_i} - x_i)\) | \([-10, 10]\) | Unimodal | 0 |
Sum power | F8 | \(f_8(x) = \sum _{i=1}^{\text {dim}} x_i^2\) | \([-1, 1]\) | Unimodal | 0 |
Sum square | F9 | \(f_9(x) = \sum _{i=1}^{\text {dim}} ix_i^2\) | \([-10, 10]\) | Unimodal | 0 |
Rosenbrock | F10 | \(f_{10}(x) = \sum _{i=1}^{\text {dim}-1} ( 100(x_{i+1} - x_i^2)^2 + (x_i - 1)^2\) | Â | Â | Â |
| Â | Â | \(- x_i^2)^2 + (x_i - 1)^2 )\) | \([-5, 10]\) | Unimodal | 0 |
Zakharov | F11 | \(f_{11}(x) = \sum _{i=1}^{\text {dim}} x_i^2 + \left( \sum _{i=1}^{\text {dim}} 0.5ix_i \right) ^2\) | Â | Â | Â |
| Â | Â | \(+ \left( \sum _{i=1}^{\text {dim}} 0.5ix_i \right) ^4\) | \([-5, 10]\) | Unimodal | 0 |
Trid | F12 | \(f_{12}(x) = \sum _{i=1}^{\text {dim}} (x_i - 1)^2 - \sum _{i=2}^{\text {dim}} x_i x_{i-1}\) | \([-5, 10]\) | Unimodal | 0 |
Elliptic | F13 | \(f_{13}(x) = \sum _{i=1}^{\text {dim}} (10^6)^{i/(\text {dim}-1)} x_i^2\) | \([-100, 100]\) | Unimodal | 0 |
Cigar | F14 | \(f_{14}(x) = x_1^2 + 10^6 \sum _{i=2}^{\text {dim}} x_i^2\) | \([-100, 100]\) | Unimodal | 0 |
Rastrigin | F15 | \(f_{15}(x) = \sum _{i=1}^{\text {dim}} \left( x_i^2 - 10 \cos (2\pi x_i) + 10 \right)\) | \([-5.12, 5.12]\) | Unimodal | 0 |
NCRastrigin | F16 | \(f_{16}(x) = \sum _{i=1}^{\text {dim}} \left( x_i^2 - 10 \cos (2\pi x_i) + 10 \right) , y_i = {\left\{ \begin{array}{ll} x_i, & \text {if } x_i \le 0.5 \\ x_i - 1, & \text {otherwise} \end{array}\right. }\) | \([-5.12, 5.12]\) | Multimodal | 0 |
Ackley | F17 | \(f_{17}(x) = 20e^{-0.2 \sqrt{\frac{1}{\text {dim}} \sum _{i=1}^{\text {dim}} x_i^2}} + e^{-1} \sum _{i=1}^{\text {dim}} \cos (2\pi x_i)\) | Â | Â | Â |
| Â | Â | \(+ 20 + e\) | \([-50, 50]\) | Multimodal | 0 |
Griewank | F18 | \(f_{18}(x) = 1 + \frac{1}{4000} \sum _{i=1}^{\text {dim}} x_i^2 - \prod _{i=1}^{\text {dim}} \cos \left( \frac{x_i}{\sqrt{i}} \right)\) | \([-600, 600]\) | Multimodal | 0 |
Alpine | F19 | \(f_{19}(x) = \sum _{i=1}^{\text {dim}} |x_i \sin (x_i) + 0.1x_i|\) | \([-10, 10]\) | Multimodal | 0 |
Penalized 1 | F20 | \(f_{20}(x) = \frac{\pi }{\text {dim}} \left\{ 10 \sin ^2(\pi y_1) + \sum _{i=1}^{\text {dim}-1} (y_i-1)^2 [1+10 \sin ^2(\pi y_{i+1})] \right.\) | Â | Â | Â |
| Â | Â | \(\left. + (y_{\text {dim}}-1)^2 \right\} + \sum _{i=1}^{\text {dim}} u(x_i, 10, 100, 4),\) | Â | Â | Â |
| Â | Â | \(y_i = 1 + \frac{x_i+1}{4}, u(x_i,a,k,m) = {\left\{ \begin{array}{ll} k(x_i - a)^m, & x_i > a \\ 0, & -a \le x_i \le a \\ k(-x_i - a)^m, & x_i < -a \end{array}\right. }\) | \([-100, 100]\) | Multimodal | 0 |
Penalized 2 | F21 | \(f_{21}(x) = 0.1 \left\{ \sin ^2(3\pi x_1) + \sum _{i=1}^{\text {dim}-1} (x_i-1)^2 [1+\sin ^2(3\pi x_{i+1})] \right.\) | Â | Â | Â |
| Â | Â | \(\left. + (x_{\text {dim}}-1)^2 [1+\sin ^2(2\pi x_{\text {dim}})] \right\} + \sum _{i=1}^{\text {dim}} u(x_i, 5, 100, 4)\) | \([-100, 100]\) | Multimodal | 0 |
Schwefel | F22 | \(f_{22}(x) = \sum _{i=1}^{\text {dim}} x_i \sin (\sqrt{|x_i|})\) | \([-100, 100]\) | Multimodal | 0 |
Lévy | F23 | \(f_{23}(x) = \sin ^2(3\pi x_1) + \sum _{i=1}^{\text {dim}} (x_i - 1)^2 [1 + \sin ^2(3\pi x_{i+1})]\) |  |  |  |
| Â | Â | \(+ (x_{\text {dim}} - 1)^2 [1 + \sin ^2(2\pi x_{\text {dim}})]\) | \([-10, 10]\) | Multimodal | 0 |
Weierstrass | F24 | \(f_{24}(x) = \sum _{i=1}^{\text {dim}} \left( \sum _{k=0}^{k_{\max }} a^k \cos (2\pi b^k (x_i+0.5)) \right)\) | Â | Â | Â |
| Â | Â | \(- \text {dim} \left( \sum _{k=0}^{k_{\max }} a^k \cos (\pi b^k) \right) , a = 0.5, b = 3, k_{\max } = 20\) | \([-1, 1]\) | Multimodal | 0 |
Solomon | F25 | \(f_{25}(x) = 1 - \cos \left( 2\pi \sqrt{\sum _{i=1}^{\text {dim}} x_i^2} \right) + 0.1 \sqrt{\sum _{i=1}^{\text {dim}} x_i^2}\) | \([-100, 100]\) | Multimodal | 0 |
Bohachevsky | F26 | \(f_{26}(x) = \sum _{i=1}^{\text {dim}} \left( x_i^2 + 2x_i^2 - 0.3\cos (3\pi x_i) \right)\) | \([-10, 10]\) | Multimodal | 0 |
