On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems

TL;DR

This paper introduces a distributed dual gradient algorithm for linearly constrained separable convex problems, proving it achieves global linear convergence under certain conditions, with applications in control, network, and power systems.

Contribution

It presents the first proof of linear convergence for dual gradient methods applied to practical linearly constrained convex problems.

Findings

Proves global linear convergence under strong convexity and Lipschitz gradient assumptions.

Develops a fully distributed dual gradient scheme with weighted step size.

Numerical simulations confirm theoretical convergence rates.

Abstract

In this paper we propose a distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipshitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation. Many real applications, e.g. distributed model predictive control, network utility maximization or optimal power flow, can be posed as linearly constrained separable convex problems for which dual gradient type methods from literature have sublinear convergence rate. In the present paper we prove for the first time that in fact we can achieve linear convergence rate for such algorithms when they are used for solving these applications. Numerical simulations are also provided to confirm our theory.