## 3 Optimization Strategies and Results  
The considered optimization regards a 4D problem that is carried out in the following mixed integer design space \(\mathbf{X}=\left(N_{t},E(D),L,R_{l}\right)\) (see Figures 1c and 2), with \(N_{t}\) being an integer and \(E(D)\) the enumeration of wire diameters. The variable \(N_{t}\) is considered continuous because its resolution is high (practical value is of order of a few hundreds) and rounding to the closest integer value has only a slight impact on power. The resolution of the second variable is low as only six different wire diameters are taken into account. Moreover, the sensitivity of power to variation in \(D\) is considerable. It should also be noticed that this variable should be converted from enumeration to an integer. The problem definition is  
maximize \(P(\mathbf{X})\)  
subject to  
\[ \begin{gathered} 200\leq X_{1}\leq 1200,\\ E(D)=\left(0.25,0.30,0.35,\cdots,0.50\right)\text{mm}\Rightarrow X_{2} =\left(1,2,3,\cdots,6\right)\in Z,\\ 0\leq X_{3}\leq 18\text{ mm},\\ 100\;\Omega\leq X_{4}\leq 8000\;\Omega,\end{gathered}  \tag{10}\]  
Due to the considerable time required for the evaluation of power using the computer model presented in Section 2, the function \(P(\mathbf{X})\) is replaced by the surrogate model. The response \(R\) of the computer model is evaluated at nodes of the Latin hypercube sampling-based design (LHS) grid \(\mathbf{X}_{\text{LHS}}:P(\mathbf{X}_{\text{LHS}})\Leftarrow R(\mathbf{X}_{ \text{LHS}})\). Finally, the surrogate model \(P(\mathbf{X})\) is created using the kriging approximation \(K:P(\mathbf{X})=K(\mathbf{X}_{\text{LHS}},R(\mathbf{X}_{\text{LHS}}))\). The three following strategies are employed for the solution of problem (10):  
**Strategy I:** Problem (10) is solved as its stands using nonlinear programming with an \"integer\" constraint equation in the form \(round(X_{2})-X_{2}=0\). This can be considered as the most _ad hoc_ approach that can easily be implemented using many existing nonlinear search algorithms. Here we use the  
Figure 3: Diagram of computer model. combination of global \(\rightarrow\) local searches using genetic [29] and the interior point algorithms [30] one after another. The choice of a genetic algorithm as a global search method results from its ability to reliably localize the region with global extremum. The interior point algorithm is applied as a local search method due to its high precision and the ability to prevent violation of inequality constraints. The strategy can be explained by the following pseudocode. \begin{tabular}{c c c c} \hline \hline \
\begin{tabular}{c} Generate LHS grid and round \(X_{2}\) \\ Perform computer experiment \\ Create 4-D surrogate model \\ Optimize 4-D surrogate model with \"integer\" constraint \\ \end{tabular} & \multicolumn{1}{c}{
\begin{tabular}{c} Generate LHS grid and round \(X_{2}\) \\ Perform computer experiment \\ Create 4-D surrogate model \\ Optimize 4-D surrogate model with \"integer\" constraint \\ \end{tabular} } \\ \hline \hline \end{tabular}   
The results obtained for different numbers of the LHS grid points are summarized in Table 1.  
**Strategy II**: Relaxation of integer variable. With only one integer variable considered, the solution strategy becomes a simplification of the branch-and-bound algorithm [21] with far fewer sequences required to find the global extremum. For the solution of each 3D subproblem, we use once again the genetic and interior point algorithms. The following pseudocode explains the strategy. \begin{tabular}{c} Relax integer variable \\ Generate 3-D LHS grid for continuous variables \\
**for each** value of integer variable \\ Create 3-D surrogate model \\ Optimize 3-D surrogate model and save solution \\ end \\ Find global extremum by ranking of extrema of relaxed subproblems \\ \end{tabular}   
The assumed resolution of the LHS grid is limited to 200 points, as shown in Table 1, to provide enough accurate localization of the extremum even for the case with four variables accounted for.