On the R-superlinear convergence of the KKT residuals generated by the augmented Lagrangian method for convex composite conic programming

• 成果类型: Article
• 署名作者: Cui, Ying; Sun, Defeng; Toh, Kim-Chuan
• 署名单位: University of Southern California; Hong Kong Polytechnic University; National University of Singapore
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1300-6
• 发表日期: 2019
• 页码: 381-415
• 关键词: error-bounds; optimization problems; local convergence; multipliers; algorithm; nondegeneracy; constraints; REGULARITY; EQUALITY; matrix

摘要

Due to the possible lack of primal-dual-type error bounds, it was not clear whether the Karush-Kuhn-Tucker (KKT) residuals of the sequence generated by the augmented Lagrangian method (ALM) for solving convex composite conic programming (CCCP) problems converge superlinearly. In this paper, we resolve this issue by establishing the R-superlinear convergence of the KKT residuals generated by the ALM under only a mild quadratic growth condition on the dual of CCCP, with easy-to-implement stopping criteria for the augmented Lagrangian subproblems. This discovery may help to explain the good numerical performance of several recently developed semismooth Newton-CG based ALM solvers for linear and convex quadratic semidefinite programming.