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

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 converge weakly to a saddle point of the Lagrangian of the convex optimization problem. The dynamical system provides through time discretization the alternating minimization algorithm AMA and also its proximal variant recently introduced in the literature.

A Dynamical Alternating Direction Multiplier Method for Two-Block Optimization Problems

“…Csetnek and others considered a Douglas-Rachford type dynamical system in [17]. And recently, a primal-dual dynamical system was introduced in [2], and a proximal alternating minimization algorithm dynamical system was proposed in [3].…”

“…In reference [2], the discrete algorithm for dynamical systems is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm. In reference [3], the discrete algorithm corresponding to the dynamical system is the proximal AMA algorithm. In reference [4], the discrete algorithms corresponding to the dynamical system are some fast inertial approximate ADMM algorithms.…”

“…In addition, in the process of solving the problem, this paper uses tools similar to literature [2] and [3]. Next, we focus on the differences between this paper and these two literatures.…”

“…Secondly, our dynamical system corresponds to different discrete algorithms, which have been mentioned above. Finally, in our dynamical system, there is an additional parameter c (it is a time variable), which is different from reference [2], and we also have another additional parameter σ, which is different from reference [3] (the motivation of taking variables c and σ comes from reference [6], where the numerical scheme ADMM also contains variable parameters c and σ).…”