Global optimization test functions

Considered in a two-dimensional input space, the unconstrained global optimization test functions in which the method was tested on included: Sphere, Rosenbrock, Rastrigin, Griewank, Goldstein-Price, Easom, and Schwefel*. The objective was to find a zone of data with maximum similarity between each test function and a quadratic function, which is presented below. The optimization problem was executed using two different initial solutions for each function, from which the best results were selected.
*For more information, please refer to [2,8].
  1. Function(s) to approximate were typical optimization test functions in literature, as previously mentioned.
  2. The function to superimpose was a metamodel, a second order polynomial linear regression of the form
\begin{equation} \begin{matrix}{\ Z=f\left(x_{1},x_{2}\right)=\beta_{0}+\ \beta_{1}x}_{1}^{2}+\ {\beta_{2}x}_{2}^{2}\ .\#\left(6\right)\\ \end{matrix}\nonumber \\ \end{equation}
The ranges of the variable bounds for this quadratic function were the same in which test function varies respectively.
To generate the grid of experimental points , the ranges of the variables of each test function were divided according to a specific value of delta x (∆x) (see Table 2).
Table 2 Step size used to generate grid of experimental points for global optimization test functions.