Simpliciality of strongly convex problems

摘要

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 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 ＞ 2). By the transversality theorem, we also give an application of singularity theory to a strongly convex C~r problem for 2 ＜ r ＜ ∞.