On Upper Approximations of Pareto Fronts

TL;DR

This paper explores conditions for the existence of upper approximations of Pareto fronts in multiobjective optimization, proposing methods to construct upper shells via problem relaxations and illustrating with a mechanical example.

Contribution

It provides sufficient conditions for upper shell existence and introduces constructive methods using problem relaxations to find upper approximations.

Findings

Sufficient conditions for upper shell existence are identified.

Constructive methods for deriving upper shells are proposed.

Numerical example demonstrates practical applicability.

Abstract

In one of our earlier works, we proposed to approximate Pareto fronts to multiobjective optimization problems by two-sided approximations, one from inside and another from outside of the feasible objective set, called, respectively, lower shell and upper shell. We worked there under the assumption that for a given problem an upper shell exists. As it is not always the case, in this paper we give some sufficient conditions for the existence of upper shells. We also investigate how to constructively search infeasible sets to derive upper shells. We approach this issue by means of problem relaxations. We formally show that under certain conditions some subsets of lower shells to relaxed multiobjective optimization problems are upper shells in the respective unrelaxed problems. Results presented are illustrated by a numerical example representing a~small but real mechanical problem. Practical implications of the results are discussed.