Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper

Proximal point algorithms on Hadamard manifolds: linear convergence and finite termination

In the present paper, we consider inexact proximal point algorithms for finding singular points of multivalued vector fields on Hadamard manifolds. The rate of convergence is shown to be linear under the mild assumption of metric subregularity. Furthermore, if the sequence of parameters associated with the iterative scheme converges to 0, then the convergence rate is superlinear. At the same time, the finite termination of the inexact proximal point algorithm is also provided under a weak sharp minima-like condition. Applications to optimization problems are provided. Some of our results are new even in Euclidean spaces, while others improve and/or extend some known results in Euclidean spaces. As a matter of fact, in the case of exact proximal point algorithm, our results improve the corresponding results in [G. C. Bento and J. X. Cruz Neto, Optim., 63 (2014), pp. 1281–1288]. Finally, several examples are provided to illustrate that our results are applicable while the corresponding results in the Hilbert space setting are not.National Natural Science Foundation of ChinaZhejiang Provincial Natural Science Foundation of ChinaDirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa