## 2 Robust Design Optimization Methods and Challenges
### Measures for Evolutionary Algorithms and Uncertainties  
The selection of the best or the most appropriate metaheuristic is a general need for electrical machine designers [5; 8; 167]. However, due to the no free lunch theorem of mathematical optimization [24; 168], there is not a metaheuristic method that is superior to any other for all of the possible problems. One necessary consequence is that the different algorithms should be benchmarked and compared on a task that is similar to the given design optimization task [23]. This problem is still not widely investigated in electrical machinery papers. There are only a few open benchmark problems, like the multiobjective TEAM (Testing Electromagnetic Analysis Methods) benchmark problems [169; 170; 171], which provide a theoretical test problem for a multiobjective and robust homogeneous magnetic field design. Moreover, benchmarking and making high-level performance measures for evolutionary and genetic algorithms are still open questions [23]. There are many mathematical benchmarks that have been produced to test and compare the different optimization algorithms. These benchmarks usually contain and mimic the following difficulties that the optimizer has to handle [38; 172]: slow convergence, a large number of local optima, a large number of variables, the dependency of variables, constraints, deceptive search spaces, flat search spaces, and uncertainties. There are different test sets to benchmark and compare the different optimization algorithms [173]. One of the most widely used is the Congress on Evolutionary Computation (CEC) competition benchmark sets [174]. Due to the lack of specific benchmark problems, these and similar general performance measures are usually used in the electrical machinery literature to select and compare the metaheuristics for a specific task, instead of using a similar, specific benchmark problem, as recommended by the no free lunch theorem of mathematical optimization [174; 175]. Due to the difficulties of handling uncertainties, there are some robust design optimization test functions published in the literature [38; 176; 177; 178; 179; 180]. Reference [38] proved the ineffectiveness of the normal PSO and genetic algorithms on these test problems. Replacing their objective functions with expectation measures, these problems can be solved with the application of the robust versions of the PSO and genetic algorithms. The solution of a robust alpine function is illustrated in Figure 1b. There are many techniques that have been introduced in the literature that can increase the performance of the evolutionary algorithm based calculations without increasing the number of samples [37; 181]. [PERSON] proposed an evolutionary algorithm based space decomposition method [182; 183], which divides the search region into high- and low-performance regions as a function of the sensitivity of the examined parameter. This technique performs further optimization only in the high-performance regions. This technique can significantly decrease the computational cost. However, as shown by [PERSON] [86], nothing guarantees that the optimal solution is in the high-performance region. Another possibility is explicit averaging over time, where explicit averaging means a form of resampling. Increasing the sample size is equivalent to reducing the variance of the estimated fitness function. [PERSON] and [PERSON] were the first to propose adapting the sample size during the run. They proposed to start the calculation with a relatively small sample size and increase the number of individuals with the number of generations [184; 185]. Other adaptation methodologies can use a higher sample size for those individuals that have a higher estimated variance or simply calculate the fitness by averaging over the neighborhood of the evaluated individuals [48]. Another possibility to reduce the noise is implicit averaging because the area of a possible solution is sampled repeatedly, and this information can be used for an implicit averaging.