Nonconvex weak sharp minima on Riemannian manifolds

TL;DR

This paper extends the theory of weak sharp minima from convex to nonconvex optimization problems on Riemannian manifolds, providing necessary conditions and applications to graph Cheeger constants.

Contribution

It generalizes characterizations of weak sharp minima to nonconvex Riemannian problems using subdifferential and cone theories, with applications to graph optimization.

Findings

Established necessary conditions for weak sharp minima on Riemannian manifolds.

Connected weak sharp minima to Cheeger constants in graph theory.

Unified various definitions of contingent cones on Riemannian manifolds.

Abstract

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.