Compositions and Averages of Two Resolvents: Relative Geometry of Fixed Points Sets and a Partial Answer to a Question by C. Byrne

TL;DR

This paper explores the geometric relationships between fixed points of resolvent averages and compositions, providing new methods to find minimizers and solutions in convex analysis and addressing a question by C. Byrne.

Contribution

It introduces a novel approach using resolvent and proximal averages to relate fixed points of resolvent compositions to minimizers of convex functions.

Findings

Fixed points of averaged resolvents relate to fixed points of compositions.

Minimizers of proximal averages can be derived from fixed points of proximal mappings.

Partial solution to C. Byrne's question on fixed points and convex sets.

Abstract

We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted considerable attention in convex analysis. Our results imply that the minimizers of proximal-average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions. When both convex functions in the proximal average are indicator functions of convex sets, least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings. This provides a partial answer to a question posed by C. Byrne. A novelty of our approach is to use the notion of resolvent average and proximal average.