Resolvent Moreau identities without monotonicity: theory and applications to Gabay duality, Douglas--Rachford and ADMM

TL;DR

This paper introduces a generalized duality for resolvent operators applicable to nonconvex problems, extending classical duality concepts and analyzing operator schemes like Douglas--Rachford and ADMM.

Contribution

It establishes a duality-like relationship for resolvent operators beyond monotonicity, extending Gabay's duality to nonmonotone problems and designing new convergent schemes.

Findings

Duality for resolvent operators extends to nonmonotone problems.

Counterexamples to ADMM convergence are provided.

New resolvent homotopy schemes effectively solve nonconvex regularized problems.

Abstract

Duality is most often defined as a relationship between convex functions. If those functions are nonconvex, classical duality breaks down. Notwithstanding, we show that another kind of duality still exists, not between the functions themselves, but between the so-called resolvent operators used to solve associated problems. In fact, this duality-like relationship holds for any set-valued mapping, and is a generalization of the Moreau's identity. We use this duality to study existing operator schemes and to design new ones. In particular, we show that the duality-like relationship Daniel Gabay illuminated between the Douglas--Rachford splitting (DR) and the Alternating Direction Method of Multipliers (ADMM) extends to nonmonotone inclusion problems. We use this relationship to provide explicit counterexamples to the convergence of ADMM in several open cases, by studying the (easier to analyse) DR scheme. Motivated by our observations, we design a class of convergent resolvent homotopy schemes and use them to solve nonconvex-regularised least absolute deviations problems. This important problem class has received little attention in the literature, since the convex component of the objective does not enjoy strong convexity.