Dynamical Primal-Dual Accelerated Method with Applications to Network Optimization

TL;DR

This paper introduces a novel continuous-time primal-dual accelerated method with optimal convergence rate for convex optimization problems with affine constraints, and applies it to network optimization tasks.

Contribution

It develops the first continuous-time primal-dual accelerated method with optimal convergence rate and applies it to distributed network optimization problems.

Findings

Achieves $O(1/t^2)$ convergence rate in duality gap.

Develops two distributed algorithms for network optimization.

Numerical simulations confirm the method's effectiveness.

Abstract

This paper develops a continuous-time primal-dual accelerated method with an increasing damping coefficient for a class of convex optimization problems with affine equality constraints. This paper analyzes critical values for parameters in the proposed method and prove that the rate of convergence in terms of the duality gap function is $O(\tfrac{1}{t^2})$ by choosing suitable parameters. As far as we know, this is the first continuous-time primal-dual accelerated method that can obtain the optimal rate. Then this work applies the proposed method to two network optimization problems, a distributed optimization problem with consensus constraints and a distributed extended monotropic optimization problem, and obtains two variant distributed algorithms. Finally, numerical simulations are given to demonstrate the efficacy of the proposed method.