Table 1 Definition of the selected unconstrained and constrained test function

\(f_{1}\) Sphere

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = \sum\limits_{i = 1}^{n} {x_{i}^{2} }\)

\(- \,5 \le x_{i} \le + 5\)

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = f\left( {0, \ldots ,0} \right) = 0\)

\(f_{2}\) Shubert

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = - \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{5} {j\sin \left[ {\left( {j + 1} \right)x_{i} + j} \right]} }\)

\(- 10 \le x_{i} \le + 10\)

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = - 24.0625\)

\(f_{3}\) Rastrigin

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = \mathop \sum \limits_{i = 1}^{n} \left( {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right)\)

\(- 5.12 \le x_{i} \le + 5.12\)

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = 0\)

\(f_{4}\) Rosenbrock

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = \mathop \sum \limits_{i = 1}^{n} \left( {100\left( {x_{x} - x_{i}^{2} } \right)^{2} + \left( {1 - x_{i} } \right)^{2} } \right)\)

\(- 2 \le x_{i} \le + 2\)

\(f\left( {x_{1} , \ldots ,x_{n} } \right) = f\left( {1, \ldots ,1} \right) = 0\)

\(f_{5}\) Easom

\(f\left( {x_{1} ,x_{2} } \right) = - \cos \left( {x_{1} } \right) \cdot \cos \left( {x_{2} } \right) \cdot {\text{e}}^{{ - \left( {\left( {x_{1} - \pi } \right)^{2} + \left( {x_{2} - \pi } \right)^{2} } \right)}}\)

\(- 100 \le x_{i} \le + 100\;i = 1,2\)

\(f\left( {x_{1} ,x_{2} } \right) = f\left( {\pi ,\pi } \right) = - 1\)