Table 3 Details of the CEC2005 benchmark functions.
Type | Definition | S | d | Fopt |
|---|---|---|---|---|
Unimodal test functions | \(f_{1} \left( X \right) = \sum\nolimits_{i = 1}^{d} {x_{i}^{2} }\) | [− 100, 100]d | 30 | 0 |
\(f_{2} \left( X \right) = \sum\nolimits_{i = 1}^{d} {\left| {x_{i} } \right|} + \prod\nolimits_{i = 1}^{d} {\left| {x_{i} } \right|}\) | [− 10, 10]d | 30 | 0 | |
\(f_{3} \left( X \right) = \sum\nolimits_{i = 1}^{d} {\left( {\sum\nolimits_{j = 1}^{i} {x_{j} } } \right)^{2} }\) | [− 100, 100]d | 30 | 0 | |
\(f_{4} \left( X \right) = max\left\{ {\left| {x_{i} } \right|,\quad 1 \le i \le d} \right\}\) | [− 100, 100]d | 30 | 0 | |
\(f_{5} \left( X \right) = \sum\nolimits_{i = 1}^{d - 1} {\left[ {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]}\) | [− 30, 30]d | 30 | 0 | |
\(f_{6} \left( X \right) = \sum\nolimits_{i = 1}^{d} {\left( {\left[ {x_{i} + 0.5} \right]} \right)^{2} }\) | [− 100, 100]d | 30 | 0 | |
\(f_{7} \left( X \right) = \sum\nolimits_{i = 1}^{d} i x_{i}^{4} + random\left[ {0, 1} \right.)\) | [− 1.28, 1.28]d | 30 | 0 | |
Multimodal test functions | \(f_{8} \left( X \right) = \sum\nolimits_{i = 1}^{d} - x_{i} \sin \left( {\sqrt {\left| {x_{i} } \right|} } \right)\) | [− 500, 500]d | 30 | − 418.982*d |
\(f_{9} \left( X \right) = \sum\nolimits_{i = 1}^{d} {\left[ {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right]}\) | [− 5.12, 5.12]d | 30 | 0 | |
\(f_{10} \left( X \right) = - 20exp\left( { - 0.2\sqrt {\frac{1}{30}\sum\nolimits_{i = 1}^{d} {x_{i}^{2} } } } \right) - exp\left( {\frac{1}{30}\sum\nolimits_{i = 1}^{d} {\cos } 2\pi x_{i} } \right) + 20 + e\) | [− 32, 32]d | 30 | 0 | |
\(f_{11} \left( x \right) = \frac{1}{4000}\sum\nolimits_{i = 1}^{d} {x_{i}^{2} } - \prod\nolimits_{i = 1}^{d} {\cos } \left( {\frac{{x_{i} }}{{\sqrt {x_{i} } }}} \right) + 1\) | [− 600, 600]d | 30 | 0 | |
\(\begin{aligned}f_{12} \left( x \right) &= \frac{\pi }{d}\left\{ 10\sin \left( {\pi y_{1} } \right) + \sum\nolimits_{i = 1}^{d - 1} {\left( {y_{1} - 1} \right)^{2} } \left[ {1 + 10sin^{2} \left( {\pi y_{i + 1} } \right)} \right]\right.\\ &\quad\left.+ \sum\nolimits_{i = 1}^{d} u \left( {x_{i} ,10,100,4} \right) \right\}\end{aligned}\) | [− 50, 50]d | 30 | 0 | |
\(\begin{aligned}f_{13} \left( x \right) &= 0.1\left\{ sin^{2} \left( {3\pi x_{i} } \right) + \sum\nolimits_{i = 1}^{d} {\left( {x_{i} - 1} \right)^{2} } \left[ {1 + sin^{2} \left( {3\pi x_{i} + 1} \right)} \right]\right.\\ &\quad\left.+\left( {x_{n} - 1} \right)^{2} \left[ {\sin^{2} \left( {2\pi x_{n} } \right)} \right] \right\} + \sum\nolimits_{i = 1}^{d} u \left( {x_{i} ,5,100,4} \right)\end{aligned}\) | [− 50, 50]d | 30 | 0 | |
Multimodal test functions with fixed dimension | \(f_{14} \left( X \right) = \left[ {\frac{1}{500} + \sum\nolimits_{j = 1}^{25} {\frac{1}{{j + \sum\nolimits_{i = 1}^{2} {\left( {x_{i} - a_{ij} } \right)^{6} } }}} } \right]^{ - 1}\) | [− 65.53, 65.53]d | 2 | 1 |
\(f_{15} \left( X \right) = \sum\nolimits_{i = 1}^{11} {\left[ {a_{i - } \frac{{x_{i} \left( {b_{i}^{2} + b_{i} x_{2} } \right)}}{{b_{i}^{2} + b_{i} x_{3} + x_{3} }}} \right]^{2} }\) | [− 5, 5]d | 4 | 0.0003 | |
\(f_{16} \left( X \right) = 4x_{1}^{2} - 2.1x_{1}^{4} + \frac{1}{3}x_{1}^{6} + x_{1} x_{2} - 4_{2}^{2} + 4_{2}^{4}\) | [− 5, 5]d | 2 | − 1.0316 | |
\(f_{17} \left( X \right) = \left( {x_{2} - \frac{5.1}{{4\pi^{2} }}x_{1}^{2} + \frac{5}{\pi }x_{1} - 6} \right)^{2} + 10\left( {1 - \frac{1}{8\pi }} \right)\cos x_{i} + 10\) | [− 5, 10]*[0, 15] | 2 | 0.398 | |
\(\begin{aligned} f_{18} \left( X \right) & = \left[ {1 + \left( {x_{1} + x_{2} + 1} \right)^{2} \left( {19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} + 6x_{1} x_{2} + 3x_{2}^{2} } \right)} \right] \\ & \quad *\left[ {30 + \left( {2x_{1} - 3x_{2} } \right)^{2} *\left( {18 - 32x_{1} + 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} } \right)} \right] \\ \end{aligned}\) | [− 5, 5]d | 2 | 3 | |
\(f_{19} \left( X \right) = - \sum\nolimits_{i = 1}^{4} {c_{i} } \exp \left[ { - \sum\nolimits_{j = 1}^{d} {a_{ij} } \left( {x_{j} - p_{ij} } \right)^{2} } \right]\) | [0, 1]d | 3 | − 3.86 | |
\(f_{20} \left( X \right) = - \sum\nolimits_{i =1}^{4} {\acute{c}}_{i} exp\left[ { - \sum\nolimits_{j =1}^{d} {\acute{a}}_{ij} \left( {x_{j} - {\acute{p}}_{ij} } \right)^{2} } \right]\) | [0, 1]d | 6 | − 3.32 | |
\(f_{21} \left( X \right) = - \sum\nolimits_{i = 1}^{5} {\left[ {\left( {x - a_{i} } \right)\left( {x - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }\) | [0, 10]d | 4 | − 10.1532 | |
\(f_{22} \left( X \right) = - \sum\nolimits_{i = 1}^{7} {\left[ {\left( {x - a_{i} } \right)\left( {x - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }\) | [0, 10]d | 4 | − 10.4028 | |
\(f_{23} \left( X \right) = - \sum\nolimits_{i = 1}^{10} {\left[ {\left( {x - a_{i} } \right)\left( {x - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} }\) | [0, 10]d | 4 | − 10.5363 |
