On Set-Based Multiobjective Optimization

Abstract

Abstract-Assuming that evolutionary multiobjective optimization (EMO) mainly deals with set problems, one can identify three core questions in this area of research: (i) how to formalize what type of Pareto set approximation is sought, (ii) how to use this information within an algorithm to efficiently search for a good Pareto set approximation, and (iii) how to compare the Pareto set approximations generated by different optimizers with respect to the formalized optimization goal. There is a vast amount of studies addressing these issues from different angles, but so far only few studies can be found that consider all questions under one roof. This paper is an attempt to summarize recent developments in the EMO field within a unifying theory of set-based multiobjective search. It discusses how preference relations on sets can be formally defined, gives examples for selected user preferences, and proposes a general, preference-independent hill climber for multiobjective optimization with theoretical convergence properties. Furthermore, it shows how to use set preference relations for statistical performance assessment and provides corresponding experimental results. The proposed methodology brings together preference articulation, algorithm design, and performance assessment under one framework and thereby opens up a new perspective on EMO.

A Hypervolume-Based Optimizer for High-Dimensional Objective Spaces

“…Different definitions can be found in the literature, and we here use the one from Zitzler et al (2008) which draws upon the Lebesgue measure as proposed in Laumanns et al (1999) and considers a reference set of objective vectors. Definition 1.…”

“…In the following, we will assume that weak Pareto dominance is the underlying preference relation, i.e., a b W, f .a/ Ä f .b/ (cf. Zitzler et al 2008). 2 A key question when tackling such a set problem is how to define the optimization criterion.…”

Automatic Design of Evolutionary Algorithms for Multi-Objective Combinatorial Optimization

“…Traditionally, mating and replacement have been defined as compositions of a fitness component, designed to favor convergence, and a diversity component, meant to keep the population spread across the objective space. Following more recent work [23], we consider general preference relations comprising three lowerlevel components: (i) a set-partitioning relation, which partitions a set of solutions in a manner consistent with Pareto dominance but cannot discriminate between nondominated solutions; (ii) a quality indicator, which assigns a quality value to solutions in a manner that does not contradict Pareto dominance (often used as a refinement of the partitions); and (iii) a diversity metric, which does not need to be consistent with Pareto dominance. All the options implemented for these low-level components are shown in Table 1.…”

“…Particularly, some of the MOEAs from the literature cannot be easily represented just by the combination of a fitness and a diversity component as proposed in ParadisEO-MOEO. Instead, following [23], we use a more general preference relation, defined as a combination of a set-partitioning criterion, a quality indicator and a diversity measure. Second, although some MOEAs claim to use an external archive, this archive is used during the evolutionary search process, either for mating or selection.…”

Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II

“…In the case of multi-objective problems, the literature proposes several Pareto-based genetic algorithms: MOGA (Multi Objective Genetic Algorithm) [39], NPGA (Niched Pareto Genetic Algorithm) [40], SPEA/SPEA-II (Strength Pareto Evolutionary Algorithm) [41], and NSGA/NSGA-II (Non-dominated Sorting Genetic Algorithm) [42]. After experimental comparison of these algorithms, the authors have decided to solve the multi-objective optimization problem through reconfiguration by using the NSGA-II algorithm with significant results (optimal solutions in short execution times).…”