On compositions of special cases of Lipschitz continuous operators

Pontus Giselsson,Walaa M Moursi

doi:10.1186/s13663-021-00709-0

Abstract

Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas–Rachford and forward–backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.

Highlights

In this paper, we assume thatX is a real Hilbert space with the inner product · | · and the induced norm ·
T is L-Lipschitz continuous if (∀(x, y) ∈ X × X) Tx – Ty ≤ L x – y, and T is nonexpansive if T is 1-Lipschitz continuous, i.e., (∀(x, y) ∈ X × X) Tx – Ty ≤ x – y
We study compositions of what we call identity-nonexpansive decompositions (I-N decompositions for short) of Lipschitz continuous operators

Summary

Introduction

X is a real Hilbert space with the inner product · | · and the induced norm ·. Averaged, conically nonexpansive, and cocoercive operators are all Lipschitz continuous operators that admit special I-N decompositions. Iterating the new averaged operator yields a sequence that converges weakly to a fixed point of the conically nonexpansive operator. These properties have been instrumental in proving convergence for the Douglas–Rachford algorithm and the forward– backward algorithm. The reflected resolvents Rγ A and Rγ B are negatively conically nonexpansive, the composition is conically nonexpansive, and a sufficient averaging gives an averaged map that converges to a fixed point when iterated. Our condition is symmetric in the individual operators and allows for one of them to be scaled conic, while the rest must be scaled averaged This is in compliance with the m = 2 case in Theorem 4.2. Nonexpansive operator classes have previously appeared in [10, 11], and illustrations of more operator classes that admit particular I-N decompositions and their compositions have appeared in [14, 24] and in early preprints of [15]

Organization and notation The remainder of this paper is organized as follows

Graphical characterizations