Hölder Metric Subregularity with Applications to Proximal Point Method

Abstract

This paper is mainly devoted to the study and applications of Hölder metric subregularity (or metric q-subregularity of order q ∈ (0, 1]) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and pointbased sufficient conditions as well as necessary conditions for q-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (firstorder) 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 for finding zeros of maximal monotone operators on Hilbert spaces.

Higher-order metric subregularity and its applications

“…Replacing d(ȳ; F(x)) by d q (ȳ; F(x)) in (1.3) as 0 < q < 1 gives us the notion of Hölder metric subregularity considered recently in [14,18,19] from different viewpoints while without its strong counterpart.…”

Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations

“…This property has found a large range of applications in different areas of variational analysis as well as in optimization: for example, in the theory of optimality conditions, in the theory of subdifferentiability, in penalty methods in mathematical programming, and in the convergence analysis of algorithms for solving equations or inclusions (see, e.g., Burke [11], Ioffe [26], Ioffe and Outrata [29], Klatte and Kummer [34], Ngai and Théra [48], Zheng and Ng [58], Li and Mordukhovich [37]). In the literature there were established several sufficient conditions in terms of certain types of coderivatives or contingent derivatives for metric subregularity of multifunctions (see, for instance, Azé [5], Li and Mordukhovich [37], Ngai and Théra [46], Ngai et al [49], Zheng and Ng [58]). Recently, Gfrerer [23] has established a first order point-based criteria for the metric subregularity of set-valued mappings.…”

Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems

“…It is worth notice that error bound results with explicit exponents are indeed important for both theory and applications since they can be used, e.g., to establish explicit convergence rates of the proximal point algorithm as demonstrated in [9], [37], [38].…”