A Dynamical Approach to Two-Block Separable Convex Optimization Problems with Linear Constraints

TL;DR

This paper introduces a dynamical systems approach to solve two-block separable convex optimization problems with linear constraints, demonstrating convergence to saddle points and deriving algorithms like AMA from the continuous model.

Contribution

It develops a novel dynamical framework for convex optimization with linear constraints, connecting continuous dynamics to existing algorithms such as AMA.

Findings

Trajectories of the dynamical system converge asymptotically to saddle points.

Discretization of the system yields the AMA and proximal algorithms.

Provides theoretical convergence guarantees for the proposed approach.

Abstract

The aim of this manuscript is to approach by means of first order differential equations/inclusions convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. Each block of the objective contains a further smooth convex function. We investigate the dynamical system proposed and prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the convex optimization problem. Time discretization of the dynamical system leads to the alternating minimization algorithm AMA and also to its proximal variant recently introduced in the literature.