# Simpliciality of strongly convex problems

arXiv: https://arxiv.org/abs/1912.09328 · HTML: https://mlsanity.com/paper/1912.09328/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2019.

## Built on by (most cited first, 2 of 2)

- [All unconstrained strongly convex problems are weakly simplicial](https://mlsanity.com/paper/2106.12704/) (2021)
- [B'ezier Flow: a Surface-wise Gradient Descent Method for Multi-objective Optimization](https://mlsanity.com/paper/2205.11099/) (2022)

## Abstract (arXiv)

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 ≤ ∞.