Power Magnetic Devices. Scott D. Sudhoff

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the stopping criterion is a check to see if a sufficient number of generations have passed. If the stopping criterion has not been met, the process repeats, starting with scaling. If the stopping criterion has been met, then an estimate of the solution x** is selected as the most fit individual of the population. This estimate can then be passed to a deterministic optimization algorithm for further refinement, yielding x*.

      Much of our focus in this chapter has focused on the single‐objective optimization of an objective or fitness function f(x). However, in the case of design problems, it is normally the case that there are multiple objectives of interest. In this section, we begin our consideration of the multi‐objective optimization problem.

      Now let us consider the question of which design is the best. Let us first compare design 5 to design 7 (again using our shorthand notation of using a 5 to designate x5). Design 7 has lower loss and lower cost than design 5, and so it is clearly better than design 5. Using similar reasoning, we can see that design 8 is better than designs {4,5,7}. Now let us compare design 8 to design 6. Design 8 has lower loss, but design 6 has lower cost. At this point, it is difficult to say which is better. This leads to the concept of dominance.

      Consider two parameter vectors xa and xb. The parameter vector xa is said to dominate xb if f(xa) is no worse than f(xb) in all objectives and f(xa) is strictly better than f(xb) in at least one objective. The statement “xa dominates xb” is equivalent to the statement that “xb is dominated by xa.” Returning to the example of the previous paragraph, we can say that design 8 dominates designs {4,5,7}.

      Consider a set of parameter vectors denoted X. The nondominated set XndX is a set of parameter vectors that are not dominated by any other member parameter vector in X. In our example, X = {1, 2, …, 8} and Xnd = {1, 6, 8}. The nondominated set can be viewed as the set of best designs.

Schematic illustration of motor performance objective space. Schematic illustration of pareto-optimal set and front.

      The goal of multi‐objective optimization is to calculate the Pareto‐optimal set and the associated Pareto‐optimal front. One may argue that in the end, only one design is chosen for a given application. This is often the case. However, to make such a decision requires knowledge of what the tradeoff between competing objectives is—that is, the Pareto‐optimal front.

      There are many approaches to calculating the Pareto‐optimal set and the Pareto‐optimal front. These approaches include the weighted sum method, the ε‐constraint method, and the weighted metric method, to name a few [10]. In each of these methods, the multi‐objective problem is converted to a single‐objective optimization, and that optimization is conducted to yield a single point on the Pareto‐optimal front. Repetition of the procedure while varying the appropriate parameters yields the Pareto‐optimal front.

      In order to illustrate this procedure, let us consider the ε‐constraint method for a system with two objectives, f1(x) and f2(x), which we wish to minimize. In order to determine the Pareto‐optimal set and front, the problem

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