# All unconstrained strongly convex problems are weakly simplicial

arXiv: https://arxiv.org/abs/2106.12704 · HTML: https://mlsanity.com/paper/2106.12704/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 24 Jun 2021.

## Builds on (most cited first, 6 of 6)

- [Multi-Objective Multi-Agent Decision Making: A Utility-based Analysis and Survey](https://mlsanity.com/paper/1909.02964/) (2019)
- [Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization](https://mlsanity.com/paper/1812.05222/) (2018)
- [Topology of Pareto sets of strongly convex problems](https://mlsanity.com/paper/1904.03615/) (2019)
- [Asymptotic Risk of Bezier Simplex Fitting](https://mlsanity.com/paper/1906.06924/) (2019)
- [Simpliciality of strongly convex problems](https://mlsanity.com/paper/1912.09328/) (2019)
- [Characterization of the equality of weak efficiency and efficiency on convex free disposal hulls](https://mlsanity.com/paper/1910.02867/) (2019)

## Built on by (most cited first, 1 of 1)

- [B'ezier Flow: a Surface-wise Gradient Descent Method for Multi-objective Optimization](https://mlsanity.com/paper/2205.11099/) (2022)

## Abstract (arXiv)

A multi-objective optimization problem is Cr weakly simplicial if there exists a Cr surjection from a simplex onto the Pareto set/front such that the image of each subsimplex is the Pareto set/front of a subproblem, where 0≤ r≤ ∞. This property is helpful to compute a parametric-surface approximation of the entire Pareto set and Pareto front. It is known that all unconstrained strongly convex Cr problems are Cr-1 weakly simplicial for 1≤ r ≤ ∞. In this paper, we show that all unconstrained strongly convex problems are C0 weakly simplicial. The usefulness of this theorem is demonstrated in a sparse modeling application: we reformulate the elastic net as a non-differentiable multi-objective strongly convex problem and approximate its Pareto set (the set of all trained models with different hyper-parameters) and Pareto front (the set of performance metrics of the trained models) by using a B'ezier simplex fitting method, which accelerates hyper-parameter search.