Fast primal-dual algorithm via dynamical system for a linearly constrained convex optimization problem

TL;DR

This paper introduces a fast primal-dual algorithm derived from a second-order dynamical system with damping, achieving accelerated convergence for linearly constrained convex optimization problems, supported by theoretical analysis and numerical experiments.

Contribution

It proposes a novel primal-dual algorithm based on a discretized second-order dynamical system with damping, achieving accelerated convergence rates.

Findings

Convergence rate of O(1/k^{lpha-1}) for objective residual and feasibility.

The algorithm outperforms traditional methods in numerical experiments.

The dynamical system analysis provides insights into the acceleration mechanism.

Abstract

By time discretization of a second-order primal-dual dynamical system with damping $\alpha/t$ where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual algorithm for a linear equality constrained convex optimization problem. Under a suitable scaling condition, we show that the proposed algorithm enjoys a fast convergence rate for the objective residual and the feasibility violation, and the decay rate can reach $\mathcal{O}(1/k^{\alpha-1})$ at the most. We also study convergence properties of the corresponding primal-dual dynamical system to better understand the acceleration scheme. Finally, we report numerical experiments to demonstrate the effectiveness of the proposed algorithm.