Simpliciality of strongly convex problems (Q821589)

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 \(r\) is a positive number. The simpliciality is an important property, which can be seen in several practical problems ranging from facility location studied already. The sparse modeling is actively developed now day. If a problem is simplicial, then it is possible to efficiently compute a parametric-surface approximation of the entire Pareto set with few samples points. In this paper, the authors give a specialized transversality theorem on generic linear perturbations of a strongly convex mapping and proof another theorem for a singularity theory to a strongly convex problem. In Section 2, are presented two examples of weakly simplicial problems and some remarks. By lemmas prepared in Section 3, it is proved in Section 4. Moreover, in Section 5, all manifolds are without boundary and assumed to have countable bases. The purpose of this section is to establish the specialized transversality theorem for generically linearly perturbed strongly convex mappings, which is an essential tool for the proof of theorem in Section 6. Section 7 is an appendix, which shows demonstrations of some lemmas used already in previous sections of the article.