The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings

TL;DR

This paper establishes bounds on the minimal displacement vectors for compositions and convex combinations of firmly nonexpansive mappings, extending previous results and demonstrating the sharpness of these bounds through examples.

Contribution

It provides new bounds on minimal displacement vectors for compositions and convex combinations of firmly nonexpansive mappings, generalizing earlier work.

Findings

Bounds on minimal displacement vectors are derived.

Results are shown to be sharp through examples.

Generalization of previous asymptotic regularity results.

Abstract

Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., the have or "almost have" fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover, we present various examples illustrating that our bounds are sharp.