A projected primal-dual splitting for solving constrained monotone inclusions

TL;DR

This paper introduces a primal-dual splitting algorithm for constrained monotone inclusions that incorporates projections onto convex sets, accelerating convergence especially under strong monotonicity and convexity assumptions.

Contribution

The paper generalizes existing primal-dual methods by integrating a projection step for a priori information, and develops accelerated schemes for strongly monotone and convex problems.

Findings

The proposed algorithm effectively incorporates a priori constraints via projections.

Accelerated schemes improve convergence under strong monotonicity and convexity.

Numerical examples demonstrate faster convergence compared to traditional methods.

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.