Approximating Pareto Set Topology by Cubic Interpolation on Bi-objective Problems

Abstract

Difficult Pareto set topology refers to multi-objective problems with geometries of the Pareto set such that neighboring optimal solutions in objective space differ in several or all variables in decision space. These problems can present a tough challenge for evolutionary multi-objective algorithms to find a good approximation of the optimal Pareto set well-distributed in decision and objective space. One important challenge optimizing these problems is to keep or restore diversity in decision space. In this work, we propose a method that learns a model of the topology of the solutions in the population by performing parametric spline interpolations for all variables in decision space. We use Catmull-Rom parametric curves as they allow us to deal with any dimension in decision space. The proposed method is appropriated for bi-objective problems since their optimal set is a one-dimensional curve according to the Karush-Kuhn-Tucker condition. Here, the proposed method is used to promote restarts from solutions generated by the model. We study the effectiveness of the proposed method coupled to NSGA-II and two variations of MOEA/D on problems with difficult Pareto set topology. These algorithms approach very differently the Pareto set. We argue and discuss their behavior and its implications for model building.