Dominance, epsilon, and hypervolume local optimal sets in multi-objective optimization, and how to tell the difference

Abstract

Local search algorithms have shown good performance for several multi-objective combinatorial optimization problems. These approaches naturally stop at a local optimal set (LO-set) under given definitions of neighborhood and preference relation among subsets of solutions, such as set-based dominance relation, hypervolume or epsilon indicator. It is an open question how LO-sets under different set preference relations relate to each other. This paper reports an in-depth experimental analysis on multi-objective nk-landscapes. Our results reveal that, whatever the preference relation, the number of LO-sets typically increases with the problem non-linearity, and decreases with the number of objectives. We observe that strict LO-sets of bounded cardinality under set-dominance are LO-sets under both epsilon and hypervolume, and that LO-sets under hypervolume are LO-sets under set-dominance, whereas LO-sets under epsilon are not. Nonetheless, LO-sets under set-dominance are more similar to LO-sets under epsilon than under hypervolume. These findings have important implications for multi-objective local search. For instance, a dominance-based approach with bounded archive gets more easily trapped and might experience difficulty to identify an LO-set under epsilon or hypervolume. On the contrary, a hypervolume-based approach is expected to perform more steps before converging to better approximations.

Particle Swarm Optimization for Cooperative Multi-Robot Task Allocation: A Multi-Objective Approach

“…Although many metrics are available to evaluate two Pareto fronts, we cannot find one that is absolutely reliable. Thus, as introduced in [36], [27], we also employ two quality indicators I H and I ǫ+ to comparing the performance of algorithms, and the values of these two indicators express the closeness to the true Pareto fronts. The values can be between zero and one, and the higher is the value of I H (or the smaller is the value of I ǫ+ ), the closer it is to the true Pareto front.…”

Peeking beyond peaks: Challenges and research potentials of continuous multimodal multi-objective optimization

“…In order to facilitate a concise wording on continuous multimodal MOO for the following sections -and maybe also as consolidation of available notions (Custódio and Madeira, 2018;Liefooghe et al, 2018c; for future discussions in the EMO domainwe provide some definitions and terminology that pick up the term of efficiency in the global and local sense, and use those to extend the current notion of optimality in the MO domain. Thereby, we specifically address multimodality in a broader sense than with a mere focus on multiple global optima (as done in Liu et al, 2018b).…”

“…Note that definitions for locally and strictly locally efficient sets are also given, for example, in Liefooghe et al (2018c) and Paquete et al (2004). However, these works are rooted in the combinatorial domain and they therefore consider different search spaces and different neighborhood relations.…”

“…1 for a visual notion of this perspective). Whereas the existence of traps due to multimodality is often inherent in discrete optimization problems (Liefooghe et al, 2018c;Paquete et al, 2004Paquete et al, , 2007aLiefooghe et al, 2018a), it is less clear in the continuous case. In fact, for continuous problems in (evolutionary) MO optimization, the properties (and challenges) of multimodality are rather transferred from SO optimization than understood in the context of MOO.…”

Maximising hypervolume and minimising ϵ-indicators using Bayesian optimisation over sets

“…Although the -indicator is also Pareto-compliant, it is weaker than hypervolume because two approximation fronts may have the same -indicator value even if one is clearly better than the other in terms of Pareto optimality. Nevertheless, there is empirical evidence that the search landscapes of hypervolume and are significantly different [18] and optimising them leads to different approximations of the Pareto front [11]. Therefore, we decided to consider both in this study.…”