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This paper is mainly devoted to the study and applications of H61der metric sub-regularity (or metric q-subregularity of order q ∈E (0, 1]) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for q-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric q-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces. [ABSTRACT FROM AUTHOR]

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 establish their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and Hadamard manifolds. A number of the results obtained in this paper are also new for the case of conventional problems in finite-dimensional Euclidean spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper. [ABSTRACT FROM AUTHOR]