Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators

Abstract

In this work we propose a new splitting technique, namely Asymmetric Forward–Backward–Adjoint splitting, for solving monotone inclusions involving three terms, a maximally monotone, a cocoercive and a bounded linear operator. Our scheme can not be recovered from existing operator splitting methods, while classical methods like Douglas–Rachford and Forward–Backward splitting are special cases of the new algorithm. Asymmetric preconditioning is the main feature of Asymmetric Forward–Backward–Adjoint splitting, that allows us to unify, extend and shed light on the connections between many seemingly unrelated primal-dual algorithms for solving structured convex optimization problems proposed in recent years. One important special case leads to a Douglas–Rachford type scheme that includes a third cocoercive operator.