2018/03/11 by M. Mahdi Karkhaneei, Karkhaneei, M. Mahdi, Nezam Mahdavi‐Amiri +1
Computer Science · Mathematics · #05C40 #49J52 #90C26 #FOS: Mathematics #Mathematical Inequalities and Applications #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1803.03943
openalex publication_date 2018/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Fréchet 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 Fréchet 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.