Table 1 Benchmark functions
Function name | Formula | Range | fmin |
|---|---|---|---|
\(\displaystyle Booth (f_1)\) | \(\displaystyle f_1({x,y}) = (x+2y-7)^2+(2x+y-5)^2\) | \(\displaystyle [-10 \;10]\) | 0 |
\(\displaystyle Easom (f_2)\) | \(\displaystyle f_2({x,y}) = -\mathrm{{cos}}(x_1)\mathrm{{cos}}(x_2) \mathrm{{exp}}(-(x - \pi )^2-(y - \pi )^2)\) | \(\displaystyle [-100 \;100]\) | − 1 |
\(\displaystyle Matyas (f_3)\) | \(\displaystyle f_3({x,y}) = 0.26(x^2+y^2) -0.48xy\) | \(\displaystyle [-10 \;10]\) | 0 |
\(\displaystyle Bohachevsky1(f_4)\) | \(\displaystyle f_4({x,y}) = x^2 + 2y^2 -0.3\mathrm{{cos}}(3\pi x)-0.4\mathrm{{cos}}(4\pi y)+0.7\) | \(\displaystyle [-100 \;100]\) | 0 |
\(\displaystyle Ackley (f_{6})\) | \(\displaystyle f_{26}({x,y}) =-20\mathrm{{exp}}{\left( -0.2 \sqrt{\frac{1}{n}\sum _{i=1}^{D} {x_{i}}^2} \right) } -\mathrm{{exp}}{\left( {\frac{1}{n}\sum _{i=1}^{D} {\mathrm{{cos}}(2\pi x_{i})}} \right) }\) +20+e | \(\displaystyle [-32 \;32]\) | 0 |
\(\displaystyle Griewank (f_{7})\) | \(\displaystyle f_{25}({x,y}) = \frac{1}{4000} {\left( \sum _{i=1}^{D} {(x_{i}-100)}^2\right) }-{\left( \prod _{i=1}^{D} {\mathrm{{cos}}(\frac{x_{i}-100}{\sqrt{i}})}\right) } +1\) | \(\displaystyle [-600 \;600]\) | 0 |
\(\displaystyle Rastrigin (f_{8})\) | \(\displaystyle f_{24}({x,y}) = {\sum _{i=1}^{D} }{({x_1}^2-10\mathrm{{cos}}{2\pi x_i}+10)}\) | \(\displaystyle [-5.12 \;5.12]\) | 0 |
\(\displaystyle Rosenbrock (f_{9})\) | \(\displaystyle f_{22}({x,y}) ={\sum _{i=1}^{D} }100{(x_{i+1}-{x_{i}}^2)}^2 {(x_1-1)}^2\) | \(\displaystyle [-30 \;30]\) | 0 |
\(\displaystyle Sum squares (f_{10})\) | \(\displaystyle f_1{8}({x,y}) = {-\sum _{i=1}^{D} {ix_i}^2}\) | \(\displaystyle [-10 \;10]\) | 0 |