On the hierarchical structure of Pareto critical sets

TL;DR

This paper reveals that the boundary of the Pareto critical set in multiobjective optimization can be characterized by subproblems with fewer objectives, aiding in solving complex many-objective problems more efficiently.

Contribution

It establishes a hierarchical structure of Pareto critical sets, enabling decomposition of many-objective problems into simpler subproblems for analysis and solution.

Findings

Boundary of Pareto critical set characterized by subproblems

Decomposition approach for many-objective optimization

Insights into Pareto set structure

Abstract

In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems considering subsets of the objective functions. If the Pareto critical set is completely described by its boundary (e.g. if we have more objective functions than dimensions in the parameter space), this can be used to solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set. This technique is especially useful for efficiently solving many-objective optimization problems by breaking them down into MOPs with a reduced number of objective functions.