Using a Factored Dual in Augmented Lagrangian Methods for Semidefinite Programming

TL;DR

This paper proposes a factorization technique to eliminate the positive semidefinite constraint in dual semidefinite programming, aiming to enhance convergence rates of augmented Lagrangian methods, supported by numerical evidence.

Contribution

It introduces a factored dual approach for augmented Lagrangian methods in semidefinite programming, improving convergence and simplifying constraint handling.

Findings

Numerical results demonstrate improved convergence rates.

The factorization simplifies dual constraint handling.

The approach enhances augmented Lagrangian methods for SDP.

Abstract

In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.