The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

• 成果类型: Article
• 署名作者: Sun, Defeng; Sun, Jie; Zhang, Liwei
• 署名单位: National University of Singapore; National University of Singapore; Dalian University of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-007-0105-9
• 发表日期: 2008
• 页码: 349-391
• 关键词: multiplier methods; optimization; constraints; STABILITY; algorithm

摘要

We analyze the rate of local convergence of the augmented Lagrangian method in nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and variational analysis on the projection operator in the symmetric matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold (c) over bar > 0.