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].
- Function(s) to approximate were typical optimization test
functions in literature, as previously mentioned.
- 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.