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

© 2017, Springer Science+Business Media New York. 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.