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