A generalized primal-dual algorithm with improved convergence condition for saddle point problems

TL;DR

This paper generalizes and improves the primal-dual algorithm for saddle point problems by enhancing convergence conditions, enabling larger step sizes, and introducing heuristics for better performance, demonstrated on the assignment problem.

Contribution

It introduces a generalized primal-dual algorithm with an improved, optimal convergence condition and a heuristic for larger step sizes in specific saddle point problems.

Findings

Enhanced convergence condition is effective and optimal.

Larger step sizes lead to improved numerical performance.

Heuristic yields significantly better results on the assignment problem.

Abstract

We generalize the well-known primal-dual algorithm proposed by Chambolle and Pock for saddle point problems, and improve the condition for ensuring its convergence. The improved convergence-guaranteeing condition is effective for the generic setting, and it is shown to be optimal. It also allows us to discern larger step sizes for the resulting subproblems, and thus provides a simple and universal way to improve numerical performance of the original primal-dual algorithm. In addition, we present a structure-exploring heuristic to further relax the convergence-guaranteeing condition for some specific saddle point problems, which could yield much larger step sizes and hence significantly better performance. Effectiveness of this heuristic is numerically illustrated by the classic assignment problem.