Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods

Abstract

In this paper we conduct local convergence analysis of the inexact Newton methods for solving the generalized equation 0 ∈ f (x) + F (x) under the assumption of Hölder strong metric subregularity, where f : X → Y is a single-valued mapping while F : X ⇒ Y is a set-valued mapping between arbitrary Banach spaces. Our work are proceeded as twofold: we first explore fully the property of Hölder strong metric subregularity by establishing a verifiable necessary and sufficient condition as well as discussing its stability under small perturbations, and secondly, with the help of aforementioned theoretical analysis, we conclude that every sequence generated by the inexact (quasi) Newton method and staying in a neighborhood of the solutionx is convergent (superlinearly) of order p(1 + q) where p is the order of Hölder strong metric subregularity imposed on the mapping f + F and q is the order of Hölder calmness property for the derivative Df while p and q complement each other as long as p(1 + q) ≥ 1.

Convergence of quasi-Newton methods for solving constrained generalized equations

“…This is a very interesting issue. In fact, as pointed out recently in [40], metric regularity fails to hold in important situations. So, a topic for further research is to establish the convergence of Quasi-Newton schemes under, for instance, Hölder-type hypotheses.…”