A Squared Smoothing Newton Method for Semidefinite Programming

• 成果类型: Article; Early Access
• 署名作者: Liang, Ling; Sun, Defeng; Toh, Kim-Chuan
• 署名单位: University System of Maryland; University of Maryland College Park; Hong Kong Polytechnic University; National University of Singapore; National University of Singapore
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0311
• 发表日期: 2024
• 关键词: Augmented Lagrangian method; constraint nondegeneracy; continuation methods; COMPLEMENTARITY; CONVERGENCE; rank; regularization; optimization; algorithms; binary

摘要

This paper proposes a squared smoothing Newton method via the Huber smoothing function for solving semidefinite programming problems (SDPs). We first study the fundamental properties of the matrix-valued mapping defined upon the Huber function. Using these results and existing ones in the literature, we then conduct rigorous convergence analysis and establish convergence properties for the proposed algorithm. In particular, we show that the proposed method is well-defined and admits global convergence. Moreover, under suitable regularity conditions, that is, the primal and dual constraint nondegenerate conditions, the proposed method is shown to have a super-linear convergence rate. To evaluate the practical performance of the algorithm, we conduct extensive numerical experiments for solving various classes of SDPs. Comparison with the state-of-the-art SDP solvers demonstrates that our method is also efficient for computing accurate solutions of SDPs.