Topology of Pareto sets of strongly convex problems

TL;DR

This paper proves that strongly convex multiobjective problems are topologically equivalent to a simplex, and shows how various practical problems can be transformed into such problems while preserving their Pareto structure.

Contribution

It establishes the simplicial nature of strongly convex problems and provides a method to ensure this property through generic linear perturbations.

Findings

Strongly convex problems are shown to be simplicial under mild conditions.

Any strongly convex problem can be perturbed to satisfy these conditions.

Practical problems like location, biological modeling, and ridge regression can be transformed into strongly convex problems.

Abstract

A multiobjective optimization problem is simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we show that strongly convex problems are simplicial under a mild assumption on the ranks of the differentials of the objective mappings. We further prove that one can make any strongly convex problem satisfy the assumption by a generic linear perturbation, provided that the dimension of the source is sufficiently larger than that of the target. We demonstrate that the location problems, a biological modeling, and the ridge regression can be reduced to multiobjective strongly convex problems via appropriate transformations preserving the Pareto ordering and the topology.