Simpliciality of strongly convex problems

Abstract

A multiobjective optimization problem is Cr simplicial if the Pareto set and the Pareto front are Cr diffeomorphic to a simplex and, under the Cr diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where 0 ≤ r ≤ ∞. In the paper titled “Topology of Pareto sets of strongly convex problems”, it has been shown that a strongly convex Cr problem is Cr-1 simplicial under a mild assumption on the ranks of the differentials of the mapping for 2 ≤ r ≤ ∞. On the other hand, in this paper, we show that a strongly convex C1 problem is C0 simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex Cr mapping (r ≥ 2). By the transversality theorem, we also give an application of singularity theory to a strongly convex Cr problem for 2 ≤ r ≤ ∞.