A dual-primal balanced augmented Lagrangian method for linearly constrained convex programming

TL;DR

This paper introduces a dual-primal balanced augmented Lagrangian method for linearly constrained convex programming, enhancing convergence and efficiency over previous methods, with demonstrated numerical performance on basis pursuit problems.

Contribution

It proposes a novel dual-primal version of the balanced ALM, improving convergence analysis and numerical efficiency for convex programming with linear constraints.

Findings

Convergence analysis conducted within variational inequalities.

Numerical experiments show improved efficiency on basis pursuit.

Method inherits advantages of the original balanced ALM.

Abstract

Most recently, He and Yuan [arXiv:2108.08554, 2021] have proposed a balanced augmented Lagrangian method (ALM) for the canonical convex programming problem with linear constraints, which advances the original ALM by balancing its subproblems and improving its implementation. In this short note, we propose a dual-primal version of the balanced ALM, which updates the new iterate via a conversely dual-primal iterative order formally. The proposed method inherits all advantages of the prototype balanced ALM, and its convergence analysis can be well conducted in the context of variational inequalities. In addition, its numerical efficiency is demonstrated by the basis pursuit problem.