## 2 Robust Design Optimization Methods and Challenges
### Robustness Methods and Measures
Figure 3: Optimization and multi-physical analysis of a reluctance machine, where (**a**) shows the flux density at the optimal shape of the machine, (**b**) depicts the von Mises stress, and (**c**) plots the distribution of the Pareto-optimal solutions. The y axes represent the average of the calculated torque, while the x axes represent the standard deviation of this torque from the average (torque ripple). Moreover, two neighboring points of the Pareto-plot can be very far from each other in the parametric space (Figure 4). The robust solution should be defined differently than in the case of a single optimization problem. Therefore, the Pareto-front of a robust optimization has four different positions compared to the original, multiobjective optimization task, as shown in Figure 5. Moreover, the computational complexity of a multiobjective optimization problem is much higher [36], because the tolerances should be checked for all of the examined objective functions. The mathematical definition of robustness is not uniform in the literature [83]. There are several definitions, but they can be categorized as expectation or variance measures [38]. This terminology was introduced initially as the first- and second-type robustness by [PERSON] [36]. Figure 4: Illustration of a Pareto-front of a multiobjective optimization task. Two solution are highlighted by yellow and green color, where the yellow is sensitive and the green robust to the changes of the parameter x. In case (**a**) sensitive and robust solutions are close to each other and (**b**) solutions are distant from each other  
Figure 5: The four possible positions of the robust front compared to the Pareto-front of the multiobjective solution: (**a**) complete Pareto front is robust, (**b**) complete Pareto front is not robust, (**c**) a part of Pareto front is not robust and (**d**) a part of Pareto front is robust  
The first kind of multiobjective robust design optimization formalism uses an expectation measure, e.g., an integration or averaging the values of the solution vector (\(\vec{x}^{*}\)) in the \(\eta\) neighborhood of \(f(\vec{x}^{*})\):  
\[\text{\emph{Minimize}}\quad f^{*}(\vec{x})=\frac{1}{|\beta_{ \varepsilon}(\vec{x})|}\int_{\vec{y}\in\beta_{\varepsilon}(\vec{x})}f(\vec{y})d \vec{y} \tag{1}\] \[x\in S \tag{2}\]  
where \(|\beta_{\varepsilon(x)}|\) represents the hypervolume of the neighborhood, as illustrated in Figure 1. Several other forms of this expectation measure were introduced in the literature (e.g., explicit averaging or worst-case solution based estimation instead of the above-defined integral) to improve the performance of the applied optimization search methodology [83; 84; 85; 86; 87; 38]. In this case, N times function evaluations should be performed around the neighborhood of the function. Due to the lack of analytical formulations for real-world electrical machine design problems, a Monte Carlo method based approach is usually used to randomly create N points around the selected solution. However, the electrical machine optimization is often numerically expensive, and this technique needs many extra function evaluations; moreover, there is a chance that the same value can be calculated more than once [36] due to the random behavior of these techniques. Advanced sampling methods (e.g., Latin-hypercube sampling) from the design of experiments [61; 88] can help to overcome this problem. Another problem with the calculation of this expectation measure is shown in Figure 1a. The continuous black line shows the values of the real objective function, and the red line shows the approximated values of the expectation measure from badly positioned and a small number of calculations. Stochastic gradient methodologies [89; 90; 91] work with the problem of how we can approximate the shape of an arbitrary, non-convex, non-linear function. This problem was firstly studied by [PERSON] and [PERSON] [92]. Their fixed-point algorithm used implicit averaging and reduced step sizes to find local extrema of noisy function evaluations.