Anderson Accelerated Primal-Dual Hybrid Gradient for solving LP

TL;DR

This paper introduces AA-PDHG, an accelerated primal-dual method for linear programming that converges globally and outperforms standard PDHG in large-scale problems, with a filtered variant ensuring stability.

Contribution

The paper proposes AA-PDHG and FAA-PDHG, novel accelerated algorithms with convergence guarantees and improved speed for solving large-scale LPs.

Findings

AA-PDHG achieves faster convergence than standard PDHG.

FAA-PDHG maintains stability through filtering techniques.

Numerical experiments demonstrate significant speedups.

Abstract

We present the Anderson Accelerated Primal-Dual Hybrid Gradient (AA-PDHG), a fixed-point-based framework designed to overcome the slow convergence of the standard PDHG method for the solution of linear programming (LP) problems. We establish the global convergence of AA-PDHG under a safeguard condition. In addition, we propose a filtered variant (FAA-PDHG) that applies angle and length filtering to preserve the uniform boundedness of the coefficient matrix, a property crucial for guaranteeing convergence. Numerical results show that both AA-PDHG and FAA-PDHG deliver significant speedups over vanilla PDHG for large-scale LP instances.