A Preconditioned Augmented Lagrangian Method for Solving Semidefinite Programming Problems

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. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: The research of K.-C. Toh is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [MOE-2019-T3-1-010]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1663 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1663 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .