The primal-dual hybrid gradient method reduces to a primal method for linearly constrained optimization problems

TL;DR

This paper demonstrates that the primal-dual hybrid gradient method can be reformulated as a purely primal algorithm for linearly constrained problems, leading to improved convergence analysis and efficiency, especially in decentralized settings.

Contribution

It shows the primal-dual hybrid gradient method reduces to a primal algorithm, enabling convergence proofs in degenerate cases and enhancing efficiency in distributed optimization.

Findings

Proves convergence even when linear systems are inconsistent

Derives improved convergence rates over existing literature

Demonstrates superior efficiency in decentralized optimization

Abstract

In this work, we show that for linearly constrained optimization problems the primal-dual hybrid gradient algorithm, analyzed by Chambolle and Pock [3], can be written as an entirely primal algorithm. This allows us to prove convergence of the iterates even in the degenerate cases when the linear system is inconsistent or when the strong duality does not hold. We also obtain new convergence rates which seem to improve existing ones in the literature. For a decentralized distributed optimization we show that the new scheme is much more efficient than the original one.