## 2 Robust Design Optimization Methods and Challenges
### Multi-Objective Optimization Techniques  
As previously shown, electrical machine optimization during a sensitivity analysis task is often considered as a single objective optimization problem. However, there are many conflicting goals in the background. These kinds of optimization methods can be handled better as constrained, single-objective problems. Several examples can be found for this kind of model and optimization techniques in the literature. Perhaps one of the most general applications in electrical machine design is solving the inverse design problems with these kinds of methods, like the preliminary design of transformers or rotating designs [9; 14; 97; 98; 99; 100; 101; 102]. The advantage of this constrained single-objective formalism is that the mathematical programming based approaches can be used for the solution of these problems, like quadratic programming [103] or geometric programming [104]. Their formalism and the built-in modern interior-point method based convex optimization solvers can quickly find the solution and prove that the found solution is the global optimum or proof that its exist [15; 105]. Moreover, these methods do not require an initial guess of the values or a reference solution to start the optimization. However, applying these methods requires an analytical model of the electrical machine, which fulfills the special requirements of the different mathematical programming tasks. This is not possible in any case, e.g., the case of a complex, finite element method based calculation. However, in real-world design tasks, the objective functions are usually not only conflicting, they are incommensurable [41; 106; 107]. Therefore, in these cases, there is no unique solution that can minimize all of the given objective functions, so the solution of the task is a set of solutions, which follows the concept of Pareto-optimality [107; 108]: a solution \(x^{*}\) is Pareto-optimal if there is no other existing solution (\(y\)) such that \(f_{i}(y)\leq f_{i}(x)\) for all \(i=1,..,m\) parameters of the solution and \(f_{i}(y)<f_{i}(x)\) for at least one parameter of the solution, if the optimization task is to search for the minimal value of the feasible solutions in the decision space. There are many metaheuristic approaches used for electrical machine design optimization [5; 8; 109; 110; 111; 112; 113; 114; 115; 116], and these methods can be categorized in many different ways. Due to their different properties, these methods can be assigned more than one category [117; 118; 119]. One of the easiest ways is to categorize them as stochastic or deterministic approaches. Those methods that do not contain any randomly changed parameters are called deterministic approaches [8]. One possible categorization of some selected techniques is depicted in Figure 7. The different stochastic approaches are very widely used in electrical machine design, because these methods can be used for gradient-free search of the solutions, and they do not need an initial solution to converge to the global optimum [120]. These kinds of population based optimization methods, like Genetic Algorithm (GA) [121; 122; 123], Particle Swarm Optimization (PSO) [124], Differential Evolution (DE) [125; 126], Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [127], Firefly Algorithm (FA) [128; 129], Cuckoo Search (CS) [130], etc., are successfully used for electrical machine design optimization [131]. Figure 6: The infeasible regions in the case of a type II robustness. These regions are calculated from the (badly) sampled points of the objective function (yellow). The main difference between these algorithms is that they use different operators and methodologies to share the information between the selected parents and offspring. From this point of view, the single objective algorithms can be categorized in the following way [23]: Gradient-Guided Moves (GGM), Random Permutation (RP), Direction Based Perturbations (DBPs), Isotropic Random Walks (IRWs), and Long-Tailed, scale-free Random Walks (LTRWs).