A Projected Primal–Dual Method 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 and convex sets. The proposed algorithm incorporates a projection step onto the a priori information sets and generalizes methods proposed in the literature for solving monotone inclusions. Moreover, under the presence of strong monotonicity, we derive an accelerated scheme inspired on the primal–dual algorithm applied to the more general context of constrained monotone inclusions. In the particular case of convex optimization, our algorithm generalizes several primal–dual optimization methods by allowing a priori information on solutions. In addition, 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.