# Nonconvex weak sharp minima on Riemannian manifolds

arXiv: https://arxiv.org/abs/1803.03943 · HTML: https://mlsanity.com/paper/1803.03943/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2018.

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

- [ARCS: Accurate Rotation and Correspondence Search](https://mlsanity.com/paper/2203.14493/) (2022)
- [Decentralized Weakly Convex Optimization Over the Stiefel Manifold](https://mlsanity.com/paper/2303.17779/) (2023)

## Abstract (arXiv)

We are to establish necessary conditions (of the primal and dual types) for the set of weak sharp minima of a nonconvex optimization problem on a Riemannian manifold. Here, we are to provide a generalization of some characterizations of weak sharp minima for convex problems on Riemannian manifold introduced by Li et al. (SIAM J. Optim., 21 (2011), pp. 1523--1560) for nonconvex problems. We use the theory of the Fr'echet and limiting subdifferentials on Riemannian manifold to give the necessary conditions of the dual type. We also consider a theory of contingent directional derivative and a notion of contingent cone on Riemannian manifold to give the necessary conditions of the primal type. Several definitions have been provided for the contingent cone on Riemannian manifold. We show that these definitions, with some modifications, are equivalent. We establish a lemma about the local behavior of a distance function. Using the lemma, we express the Fr'echet subdifferential (contingent directional derivative) of a distance function on a Riemannian manifold in terms of normal cones (contingent cones), to establish the necessary conditions. As an application, we show how one can use weak sharp minima property to model a Cheeger type constant of a graph as an optimization problem on a Stiefel manifold.