Simpliciality of strongly convex problems

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 ≤ 𝑟 ≤ ∞. 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 ≤ 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 𝐶^𝑟 mapping (𝑟 ≥ 2). By the transversality theorem, we also give an application of singularity theory to a strongly convex 𝐶^𝑟 problem for 2 ≤ 𝑟 ≤ ∞.