A preconditioned augmented Lagrangian method for solving semidefinite programming problems

TL;DR

This paper introduces a preconditioned augmented Lagrangian method for semidefinite programming that accelerates convergence, especially on ill-conditioned problems, and extends to a solver for nonlinear convex SDPs.

Contribution

It proposes a novel preconditioning technique within ALM for SDPs and develops SDPF+ for large-scale, possibly nonlinear, convex problems.

Findings

Significantly accelerates ALM for ill-conditioned SDPs

Outperforms existing solvers on large-scale SDPs with low-rank solutions

Demonstrates robustness and efficiency through extensive experiments

Abstract

In this work, we propose a preconditioned augmented Lagrangian method (ALM) for solving semidefinite programming (SDP) problems. The preconditioner is implemented via a weighted penalty function in the ALM subproblem, with the weight matrix derived from the projection operator onto the tangent space of the feasible region. This simple yet effective modification significantly accelerates ALM, particularly for ill-conditioned SDPs. By combining the preconditioned ALM with our previously developed feasible method SDPF, we develop SDPF+, an SDP solver capable of handling convex problems with possibly nonlinear objective functions. Extensive numerical experiments demonstrate the efficiency and robustness of SDPF+, showing that it can generally outperform other solvers on large-scale SDPs whose optimal solutions exhibit low-rank structure.