A quadratically convergent semismooth Newton method for nonlinear semidefinite programming without generalized Jacobian regularity

• 成果类型: Article; Early Access
• 署名作者: Feng, Fuxiaoyue; Ding, Chao; Li, Xudong
• 署名单位: Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS; Fudan University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02198-0
• 发表日期: 2025
• 关键词: Augmented Lagrangian method; constraint nondegeneracy; error-bounds; algorithms; inverse

摘要

We introduce a quadratically convergent semismooth Newton method for nonlinear semidefinite programming that eliminates the need for the generalized Jacobian regularity, a common yet stringent requirement in existing approaches. Our strategy involves identifying a single nonsingular element within the Bouligand generalized Jacobian, thus avoiding the standard requirement for nonsingularity across the entire generalized Jacobian set, which is often too restrictive for practical applications. The theoretical framework is supported by introducing the weak second order condition (W-SOC) and the weak strict Robinson constraint qualification (W-SRCQ). These conditions not only guarantee the existence of a nonsingular element in the generalized Jacobian but also forge a primal-dual connection in linearly constrained convex quadratic programming. The theoretical advancements further lay the foundation for the algorithmic design of a novel semismooth Newton method, which integrates a correction step to address degenerate issues. Particularly, this correction step ensures the local convergence as well as a superlinear/quadratic convergence rate of the proposed method. Preliminary numerical experiments corroborate our theoretical findings and underscore the practical effectiveness of our method.