Computing the resolvent of the sum of operators with application to best approximation problems

TL;DR

This paper introduces a flexible splitting method for computing the resolvent of the sum of weakly monotone operators in Hilbert spaces, with guarantees of strong and linear convergence, and applies it to convex optimization problems.

Contribution

It presents a novel splitting approach for resolvent computation with convergence guarantees and applies it to best approximation and proximity operator problems.

Findings

Strong convergence of the proposed method is guaranteed.

Linear convergence is achieved under Lipschitz continuity.

Effective application to convex set intersection problems.

Abstract

We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under Lipschitz continuity assumption. The approach is then applied to computing the proximity operator of the sum of weakly convex functions, and particularly to finding the best approximation to the intersection of convex sets.