Figure 3 Example of Window of Maximal Similarity found between Sphere
and second order polynomial regression.
In Figure 3, an example is presented to demonstrate the application of
the proposed method, where function to approximate,\(\mathbf{Y\ =\ f}\left(\mathbf{R}\right)\), is the sphere and the
function to superimpose,\(\mathbf{Z\ =\ f}\left(\mathbf{R}\right)\), is a second order
polynomial 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(3\right)\\
\end{matrix}\nonumber \\
\end{equation}Since the function to superimpose is a metamodel, the optimization model
had to additionally find metamodel parameters\(\mathbf{\beta}_{\mathbf{0}}\mathbf{,}\mathbf{\beta}_{\mathbf{1}}\mathbf{,\ldots\ }\mathbf{\beta}_{\mathbf{n}}\).
For the special case where the two functions have the exact same shape,
as was the case presented for Figure 3, the problem will evidently have
infinitely many solutions since the window of maximum similarity could
adjust itself in any range in the experimental region.
The logarithm base 10, which is not defined for 0 or negative values,
was used in the objective formulation to keep in the same order of
magnitude between the SSE and the distance between bounds. Distances
between bounds
(\(\mathbf{x}_{\mathbf{i}}^{\mathbf{U}}\mathbf{-}\mathbf{x}_{\mathbf{i}}^{\mathbf{L}}\mathbf{)}\)are present:
- in the objective function in order to avoid window size dependency on
parameters
- in the constraints because a minimum value had to be included for the
formulation to be successful and not contain a single point. This
suggested value,\(\mathbf{\varepsilon=1\times}\mathbf{10}^{\mathbf{-6}}\), is
known as the non-archimedean constant, a value commonly used for
computational purposes.
Additional constraints include a range where to define the window’s
upper and lower bounds, in accordance to the span in which each variable
varies.
An important quality of the method is its use of computational
resources; all the optimization problems included in this work were
solved using Excel Solver, a local optimizer included in MS Excel. MS
Solver uses the Generalized Reduced Gradient (GRG) algorithm to solve
non-linear optimization problems and the Branch and Bound method to
solve mixed-integer and constraint programming problems [7].