Many-objective optimization and hypervolume based search

TL;DR: This thesis tackles such many-objective optimization problems in terms of theoretical investigations to better understand why classical MOEAs have difficulties with many objectives and develops objective reduction techniques that automatically reduce the number of objectives while the Pareto dominance relation is preserved or only slightly changed.

Abstract: Multiobjective optimization problems occur frequently in practice where multiple objectives have to be optimized simultaneously and the goal is to find or approximate the set of Pareto-optimal solutions. Multiobjective evolutionary algorithms (MOEAs) are one type of randomized search heuristics that are well-suited for multiobjective optimization problems due to their ability of computing a set of trade-off solutions in one run. However, current state-of-the-art MOEAs are known to have difficulties if the number of objectives is high, i.e., larger than 4. This thesis tackles such many-objective optimization problems in terms of theoretical investigations to better understand why classical MOEAs have difficulties with many objectives. New approaches and techniques are provided that enhance the search capabilities of MOEAs for many-objective problems. Furthermore, we investigate hypervolume-based MOEAs in more depth that have been proposed recently especially for many-objective optimization scenarios. In particular, we generally investigate the question of what happens if objectives are added to a problem formulation—both with respect to the Pareto dominance relation in general and with respect to the running time of specific MOEAs. This includes the proposition of a general measure to quantify the changes occurring in the Pareto dominance relation if the objective set is changed. Based on this measure, we propose the term of δ-non-conflicting objectives: if two objective sets are δ-non-conflicting with each other, the two sets can be interchangeably optimized without changing the resulting Pareto set approximation by an additional term of δ in each objective, i.e., we make an error of at most δ. Based on these theoretical foundations, objective reduction techniques are developed that automatically reduce the number of objectives while the Pareto dominance relation is preserved or only slightly changed. While reducing the number of objectives, we distinguish between two problems. We either predefine a maximal size of the sought objective subset or a maximal δ-error allowed and then ask for the objective subset of the original objectives that meets these goals and which optimizes the above mentioned δ-error and the size of the resulting objective set respectively. Both exact algorithms and greedy heuristics are proposed and analyzed with respect to their running time for each of the two problems. Furthermore, the NP-hardness of both problems is proved. Besides objective reduction by omitting objectives, we also consider the aggregation of objectives and investigate what happens with respect to the Pareto dominance relation if objectives are aggregated. Algorithms that aggregate objectives automatically are proposed and compared to their objective omission counterparts. The usefulness of the objective reduction algorithms for a decision maker in an a posteriori scenario, i.e., after the search, is presented for several test problems and a real-world application of radar waveform optimization. In the last part of the thesis, we investigate several aspects of hypervolume-based MOEAs. The hypervolume indicator has become popular to guide the search of MOEAs in the past years due to its properties of being a refinement of the Pareto dominance relation. This allows for a better guidance towards the Pareto-optimal solutions which is especially useful if many objectives are to be optimized. However, the running time for computing the hypervolume indicator exactly is exponential in the number of objectives which results in the need for further research in this area to provide more powerful and better applicable MOEAs if many objectives are to be optimized. The first contribution of this thesis part is the first rigorous running time analysis of a hypervolume-based MOEA showing that the approach of optimizing the hypervolume with a (μ+1)-strategy can find the set of Pareto-optimal solutions for a specific problem in a reasonable time. To increase the applicability of hypervolume-based MOEAs to realworld problems, two further studies are presented. First, we generalize the hypervolume indicator to a weighted version and show how this generalized indicator allows to incorporate user preferences into the search. Second, we apply the developed objective reduction algorithms within a hypervolumebased MOEA which is shown to reduce the running time needed for the hypervolume calculation and which at the same time produces solutions of higher quality.