Simpliciality of strongly convex problems

TL;DR

This paper investigates the topological structure of Pareto sets in strongly convex multiobjective problems, showing new results on their simpliciality and applying singularity theory to understand their geometric properties.

Contribution

It extends previous work by proving that strongly convex $C^1$ problems are $C^0$ simplicial and develops a transversality theorem for generic linear perturbations of strongly convex mappings.

Findings

Strongly convex $C^1$ problems are $C^0$ simplicial under mild assumptions.

A specialized transversality theorem for linear perturbations of strongly convex mappings.

Application of singularity theory to analyze the structure of Pareto sets.

Abstract

A multiobjective optimization problem is $C^r$ simplicial if the Pareto set and the Pareto front are $C^r$ diffeomorphic to a simplex and, under the $C^r$ diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where $0\leq r\leq \infty$. In the paper titled "Topology of Pareto sets of strongly convex problems," it has been shown that a strongly convex $C^r$ problem is $C^{r-1}$ simplicial under a mild assumption on the ranks of the differentials of the mapping for $2\leq r \leq \infty$. On the other hand, in this paper, we show that a strongly convex $C^1$ problem is $C^0$ simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex $C^r$ mapping $(r\geq 2)$. By the transversality theorem, we also give an application of singularity theory to a strongly convex $C^r$ problem for $2\leq r \leq \infty$.