Variation Rate to Maintain Diversity in Decision Space within Multi-Objective Evolutionary Algorithms

Oliver Cuate,Oliver Schütze

doi:10.3390/mca24030082

Abstract

The performance of a multi-objective evolutionary algorithm (MOEA) is in most cases measured in terms of the populations’ approximation quality in objective space. As a consequence, most MOEAs focus on such approximations while neglecting the distribution of the individuals of their populations in decision space. This, however, represents a potential shortcoming in certain applications as in many cases one can obtain the same or very similar qualities (measured in objective space) in several ways (measured in decision space). Hence, a high diversity in decision space may represent valuable information for the decision maker for the realization of a given project. In this paper, we propose the Variation Rate, a heuristic selection strategy that aims to maintain diversity both in decision and objective space. The core of this strategy is the proper combination of the averaged distance applied in variable space together with the diversity mechanism in objective space that is used within a chosen MOEA. To show the applicability of the method, we propose the resulting selection strategies for some of the most representative state-of-the-art MOEAs and show numerical results on several benchmark problems. The results demonstrate that the consideration of the Variation Rate can greatly enhance the diversity in decision space for all considered algorithms and problems without a significant loss in the approximation qualities in objective space.

Highlights

In many areas such as Economy, Finance, or Industry, the problem arises naturally that several conflicting objectives have to be optimized concurrently [1,2]
We have addressed the problem of computing diverse solutions both in decision and objective space for a given multi-objective optimization problem via specialized evolutionary strategies
While so far quite a few good diversity mechanisms exist to obtain a spread in objective space, the consideration of the Pareto set approximations has been mainly neglected so far

Summary

Introduction

In many areas such as Economy, Finance, or Industry, the problem arises naturally that several conflicting objectives have to be optimized concurrently [1,2]. The decision maker may select one “optimal” realization of the project, and keep solutions with similar objective values but later launch dates as back-up solutions For such problems, the sole consideration of the approximation quality in objective space represents a potential shortcoming. By each added objective, the dimension of the Pareto set/front increases by one In this context, there exists an additional challenge in solving a given MOP, since we have to find a suitable approximation to the optimal set both in objective and decision space, in order to provide a satisfying overview of the possible solutions to the decision maker. I.e., under certain mild smoothness assumption on the model, both Pareto set and front form at least locally (k − 1)-dimensional objects

Objectives

Results

Conclusion