Topology of pareto sets of strongly convex problems

Naoki Hamada, Kenta Hayano, Shunsuke Ichiki, Yutaro Kabata, Hiroshi Teramoto

Research output: Contribution to journalArticlepeer-review

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 that treats a subset of objective functions. 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.

Original languageEnglish
Pages (from-to)2659-2686
Number of pages28
JournalSIAM Journal on Optimization
Volume30
Issue number3
DOIs
Publication statusPublished - 2020

Keywords

  • Multiobjective optimization
  • Simplicial problem
  • Strongly convex mapping
  • Topology of differentiable mapping

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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