## 4 Application to Power Electronics
### Results of the Optimization Problem
#### 4.6.2 Computational Cost Aspect  
The performance of the differential evolution and SQP algorithms are compared in Table 5. The given performances correspond to one optimization run.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimized Outputs** & **Initial Point (SQP)** & \begin{tabular}{c} **Optimal Point (SQP** \\ **Exact Gradients)** \\ \end{tabular} & \
\begin{tabular}{c} **Optimal Point (SQP** \\ **Finite Differences)** \\ \end{tabular} &
\begin{tabular}{c} **Optimal Point** \\ **(Differential Evolution)** \\ \end{tabular} \\ \hline Mass of the aircraft & & & & \\ power channel (kg) & 138.5 & 41.78 & 95.57 & 39.28 \\ Routh Criteria & 729 (\(>\)0) & 1408 (\(>\)0) & 975 (\(>\)0) & 23 (\(>\)0) \\ Transient-state & 0 (\(=\)0) & 0 (\(=\)0) & 0 (\(=\)0) & 0 (\(=\)0) \\ constraint & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Optimal outputs of the optimization problem for each algorithm.  
Figure 15: Frequency-domain constraints for the optimal solutions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Criteria** & **SQP (Exact Gradients)** & **SQP (Finite Differences)** & **Differential Evolution** \\ \hline Number of optimization iterations & & & \\ (one optimization run) & 31 & 13 & 195 \\ Number of model evaluations & & & \\ (one optimization run) & 37 & 149 & 19.985 \\ Number of model gradients & & & \\ evaluations (one optimization run) & 30 & 13 & / \\ Computation time (one & 6 min & 1 h 13 min & 2 days 20 h 57 min \(\approx\) 70 h \\ optimization run) & 6 min \(\times\) 58 & & 70 h \(\times\) 3 \\ Computation time (with & 6 min \(\times\) 58 & / & **=210 h** \\ \textless{}multi-start\textgreater{} procedure) & = **6 h** & / & **=210 h** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of optimization algorithm performance.  
SQP demonstrates a remarkable speed advantage over differential evolution in a single optimization run, achieving a ratio of approximately 700. This underscores the significant advantages that gradients offer in the realm of optimization. Furthermore, the gradient computation technique employed in the article, a sophisticated blend of symbolic derivation and automatic differentiation, yields solutions that are not only more precise, but are also more swift than those derived through the use of the finite difference method.  
In this context, the optimal solution obtained from differential evolution served as the initial point for SQP, which regarded it as an optimal solution, ultimately failing to identify a superior alternative.  
The methodology articulated in reference [11], which incorporates an analytical model for frequency-domain features, alongside the SQP approach, successfully arrived at an optimal solution within a mere two hours for a single optimization run. In contrast, the innovative methodology presented in the paper resolved the optimization problem in approximately six minutes, achieving a remarkable ratio of 20 for one optimization run. This advancement signifies a notable enhancement in the optimization performance, effectively eliminating the need for a priori assumptions regarding operating modes. Additionally, the extraction of transient-state features, as introduced in this paper, represents a significant innovation, as existing methods [11] necessitate computation only after the optimization process has concluded.