Moduli of regularity and rates of convergence for Fejér monotone sequences

Abstract

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the finite termination, for classes of Fejér monotone sequences which appear in fixed point theory, monotone operator theory, and convex optimization. This concept allows for a unified approach to several notions such as weak sharp minima, error bounds, metric subregularity, Hölder regularity, etc., as well as to obtain rates of convergence for Picard iterates, the Mann algorithm, the proximal point algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative version of the well-known fact that for a convex lower semi-continuous function the set of minimizers coincides with the set of zeros of its subdifferential and the set of fixed points of its resolvent.MSC: 41A25; 41A52; 41A65; 53C23; 03F10

An abstract proximal point algorithm

“…As mentioned in [2] (see also [36]), if X is a CAT(0) space, every firmly nonexpansive mapping T : X → X satisfies the so-called property (P 2 ). Namely,…”

“…So far, proof mining has been successfully applied to obtain quantitative versions of celebrated results in various areas of mathematics such as approximation theory, nonlinear analysis, metric fixed point theory, ergodic theory, or topological dynamics. Recently, its methods have begun to be applied to convex optimization, for more details see [5,34,35,36,38,41,42].…”

Proof-Theoretic Methods in Nonlinear Analysis

“…[56,65] provide interesting instances of such a hybrid approach. In very special, but important, cases for applications one can extract rates of convergence for iterative procedures towards some non-unique zero of F or fixed point of T, namely when one has an effective so-called modulus of regularity which is closely related to the concepts of weak sharp minima and metric regularity used in convex optimization (see [48]).…”

“…This e.g. is used essentially in [4] and [48]. In the rest of the paper we give a few typical examples of explicit bounds which have been obtained by the proof-theoretic machinery discussed so far.…”

“…• [46,48,66] rates of asymptotic regularity, strong convergence (in special cases) resp. metastability for the proximal point algorithm.…”