# A primal-dual dynamical approach to structured convex minimization problems

## TL;DR

This paper introduces a primal-dual dynamical system for structured convex minimization, proving convergence to saddle points and deriving convergence rates, leading to a new numerical algorithm combining proximal methods.

## Contribution

It presents a novel dynamical system approach for structured convex problems and derives an explicit discretization that results in an effective numerical algorithm.

## Key findings

- Trajectories asymptotically converge to saddle points.
- Convergence rates for feasibility violation and objective function.
- Discretization yields a new algorithm combining proximal methods.

## Abstract

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm.