A projected primal-dual splitting for solving constrained monotone inclusions

Abstract: In this paper we provide an algorithm for solving constrained composite primal-dual monotone inclusions, i.e., monotone inclusions in which a priori information on primal-dual solutions is represented via closed convex sets. The proposed algorithm incorporates a projection step onto the a priori information sets and generalizes the method proposed in [V~u, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667-681 (2013)]. Moreover, under the presence of strong monotonicity, we derive an accelerated scheme inspired on [Chambolle, A.; Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120-145 (2011)] applied to the more general context of constrained monotone inclusions. In the particular case of convex optimization, our algorithm generalizes the methods proposed in [Condat, L.: A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158,460-479 (2013)] allowing a priori information on solutions and we provide an accelerated scheme under strong convexity. An application of our approach with a priori information is constrained convex optimization problems, in which available primal-dual methods impose constraints via Lagrange multiplier updates, usually leading to slow algorithms with unfeasible primal iterates. The proposed modification forces primal iterates to satisfy a selection of constraints onto which we can project, obtaining a faster method as numerical examples exhibit. The obtained results extend and improve several results in the literature.