Nonconvex weak sharp minima on Riemannian manifolds

M. Mahdi Karkhaneei, Nezam Mahdavi-Amiri

Published 2018-03-11Version 1

We present some 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 generalization of some characterizations of weak sharp minima for convex problems on Riemannian manifolds introduced by Li et al. (SIAM J. Optim., 21 (2011), pp. 1523-1560) for nonconvex problems. We use the theory of the Frechet and limiting subdifferentials on Riemannian manifolds 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 manifolds to give the necessary conditions of the primal type. Several definitions have been provided for the contingent cone on Riemannian manifolds. We show that these definitions, with some modifications, are equivalent. A key tool for obtaining the necessary conditions is expressing the Frechet subdifferential (contingent directional derivative) of the distance functions on the Riemannian manifolds in terms of normal cones (contingent cones); these, of course, are well-known results in linear space settings and known results in Riemannian manifolds with non-positive sectional curvature for convex problems.