Topology of Pareto sets of 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.
Keywords
Cite
@article{arxiv.1904.03615,
title = {Topology of Pareto sets of strongly convex problems},
author = {Naoki Hamada and Kenta Hayano and Shunsuke Ichiki and Yutaro Kabata and Hiroshi Teramoto},
journal= {arXiv preprint arXiv:1904.03615},
year = {2019}
}
Comments
21 pages. Remarks 4.4 and 4.5 are added. A new application is given in section 5.3. Introduction is also revised accordingly