Preconditioned primal-dual dynamics in convex optimization: non-ergodic convergence rates

TL;DR

This paper introduces a preconditioned primal-dual dynamical system for convex optimization, demonstrating non-ergodic convergence rates and improved asymptotic properties, supported by numerical experiments.

Contribution

It proposes a family of time-dependent preconditioners for primal-dual flows, establishing convergence rates and optimality results in convex optimization.

Findings

Preconditioned flows achieve fast convergence rates.

Antisymmetric preconditioners improve asymptotic behavior.

Numerical examples confirm theoretical advantages.

Abstract

We introduce and analyze a continuous primal-dual dynamical system in the context of the minimization problem $f(x)+g(Ax)$, where $f$ and $g$ are convex functions and $A$ is a linear operator. In this setting, the trajectories of the Arrow-Hurwicz continuous flow may not converge, accumulating at points that are not solutions. Our proposal is inspired by the primal-dual algorithm of Chambolle and Pock (2011), where convergence and splitting on the primal-dual variable are ensured by adequately preconditioning the proximal-point algorithm. We consider a family of preconditioners, which are allowed to depend on time and on the operator $A$, but not on the functions $f$ and $g$, and analyze asymptotic properties of the corresponding preconditioned flow. Fast convergence rates for the primal-dual gap and optimality of its (weak) limit points are obtained, in the general case, for asymptotically antisymmetric preconditioners, and, in the case of linearly constrained optimization problems, under milder hypotheses. Numerical examples support our theoretical findings, especially in favor of the antisymmetric preconditioners.