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].
- The experimental region is a grid that contains 121 points;\(\mathbf{x}_{\mathbf{1}}\mathbf{\in}\left[\mathbf{5,5}\right]\mathbf{,\ }\mathbf{x}_{\mathbf{2}}\mathbf{\in}\left[\mathbf{5,5}\right]\mathbf{\text{.\ }}\)
- The optimization problem formulation is given by (2).
- 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:
- The use of multiple starting points using a population size of 100.
- A constraint precision of\(\mathbf{1\times}\mathbf{10}^{\mathbf{-7}}\).
- A convergence of \(\mathbf{1\times}\mathbf{10}^{\mathbf{-4}}\).