Abstract. Many-objective optimization is a hot issue in the EMO (evolutionary multiobjective optimization) community. Since almost all solutions in the current population are non-dominated with each other in many-objective EMO algorithms, we may need a different fitness evaluation scheme from the case of two and three objectives. One difficulty in the design of many-objective EMO algorithms is that we cannot visually observe the behavior of multiobjective evolution in the objective space with four or more objectives. In this paper, we propose the use of many-objective test problems in a two- or three-dimensional decision space to visually examine the behavior of multiobjective evolution. Such a visual examination helps us to understand the characteristic features of EMO algorithms for many-objective optimization. Good understanding of existing EMO algorithms may facilitates their modification and the development of new EMO algorithms for many-objective optimization.

The use of Pareto dominance for fitness evaluation has been the mainstream in evolutionary multiobjective optimization for the last two decades. Recently, it has been pointed out in some studies that Pareto dominance-based algorithms do not always work well on multiobjective problems with many objectives. Scalarizing function-based fitness evaluation is a promising alternative to Pareto dominance especially for the case of many objectives. A representative scalarizing function-based algorithm is MOEA/D (multiobjective evolutionary algorithm based on decomposition) of Zhang & Li (2007). Its high search ability has already been shown for various problems. One important implementation issue of MOEA/D is a choice of a scalarizing function because its search ability strongly depends on this choice. It is, however, not easy to choose an appropriate scalarizing function for each multiobjective problem. In this paper, we propose an idea of using different types of scalarizing functions simultaneously. For example, both the weighted Tchebycheff (Chebyshev) and the weighted sum are used for fitness evaluation. We examine two methods for implementing our idea. One is to use multiple grids of weight vectors and the other is to assign a different scalarizing function alternately to each weight vector in a single grid.

In recent years, indicator-based evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of objective functions simultaneously to the singleobjective optimization goal of finding a set of µ points that maximizes the underlying indicator. Understanding the difference between these two optimization goals is fundamental when applying indicator-based algorithms in practice. On the one hand, a characterization of the inherent optimization goal of different indicators allows the user to choose the indicator that meets her preferences. On the other hand, knowledge about the sets of µ points with optimal indicator values—so-called optimal µ-distributions—can be used in performance assessment whenever the indicator is used as a performance criterion. However, theoretical studies on indicator-based optimization are sparse. One of the most popular indicators is the weighted hypervolume indicator. It allows to guide the search towards user-defined objective space regions and at the same time has the property of being a refinement of the Pareto dominance relation with the result that maximizing the indicator results in Pareto-optimal solutions only. In previous work, we theoretically investigated the unweighted hypervolume indicator in terms of a characterization of optimal µ-distributions and the influence of the hypervolume’s reference point for general bi-objective optimization problems. In this

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