# Topology of Pareto sets of strongly convex problems

arXiv: https://arxiv.org/abs/1904.03615 · HTML: https://mlsanity.com/paper/1904.03615/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2019.

## Built on by (most cited first, 4 of 4)

- [Optimization on Pareto sets: On a theory of multi-objective optimization](https://mlsanity.com/paper/2308.02145/) (2023)
- [All unconstrained strongly convex problems are weakly simplicial](https://mlsanity.com/paper/2106.12704/) (2021)
- [Approximate Bayesian Computation of B'ezier Simplices](https://mlsanity.com/paper/2104.04679/) (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 simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we show that strongly convex problems are simplicial under a mild assumption on the ranks of the differentials of the objective mappings. We further prove that one can make any strongly convex problem satisfy the assumption by a generic linear perturbation, provided that the dimension of the source is sufficiently larger than that of the target. We demonstrate that the location problems, a biological modeling, and the ridge regression can be reduced to multiobjective strongly convex problems via appropriate transformations preserving the Pareto ordering and the topology.