method evaluation

Two scenarios for evaluation of the method were considered. First, the evaluation of the method using a function to approximate which was designed in our research group, AOG_1, is presented. Lastly, a case of an application of the proposed method using unconstrained global optimization test functions is presented. The figures were generated in QtiPlot software (version 0.9.8.9) (http://www.qtiplot.com/).

AOG_1

It was of our interest to find the maximum similarity between functionAOG_1 and a quadratic function with the form of a bowl because it was reasonable to understand that, potentially, the resulting WMS will match the curve region of the function AOG_1 with the quadratic function.
The function to approximate, AOG_1 , is a piece-wise function which mostly has the form of a plane except for a given interval in the central experimental region where it looks like a bowl (as shown in Figure 3). The ranges \(\mathbf{x}_{\mathbf{1}}\) and\(\mathbf{x}_{\mathbf{2}}\) were [-5, 5]. AOG_1 is given by:
\begin{equation} \begin{matrix}Y=f\left(x_{1},x_{2}\right)=\ \left\{\begin{matrix}{\ \ 5x}_{1}^{2}+\ {5x}_{2}^{2}\text{\ \ \ \ if}\text{\ \ x}_{1}\in\left[-2,\ 1\right]\ ,\ x_{2}\in\left[-3,\ 0\ \right]\\ \\ 500-5x_{1}+{\ \ 5x}_{2}\text{\ \ for\ any\ other\ point\ }\\ \end{matrix}\right.\ \#\left(4\right)\\ \end{matrix}\nonumber \\ \end{equation}
The function to superimpose was fixed for this case and given by
\begin{equation} \begin{matrix}{\mathbf{\ Z=f}\left(\mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}\right)\mathbf{=}\mathbf{\ 5}\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+\ }{\mathbf{5}\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{\ \#}\left(\mathbf{5}\right)\\ \end{matrix}\nonumber \\ \end{equation}
The ranges \(\mathbf{x}_{\mathbf{1}}\) and\(\mathbf{x}_{\mathbf{2}}\)were also [-5, 5].
  1. The experimental region is a grid that contains 121 points;\(\mathbf{x}_{\mathbf{1}}\mathbf{\in}\left[\mathbf{5,5}\right]\mathbf{,\ }\mathbf{x}_{\mathbf{2}}\mathbf{\in}\left[\mathbf{5,5}\right]\mathbf{\text{.\ }}\)
  2. The optimization problem formulation is given by (2).
  3. The last step is to optimize the model. The global minimum forAOG_1 is given by solution (0, 0), with a corresponding objective value of 0. Essentially, the maximum similarity would be found if the WMS was adjusted within the quadratic region of both functions. The parameters for setting up the Solver that were used for this evaluation include: