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

Abstract

Due to the possible lack of primal-dual-type error bounds, the superlinear convergence for the Karush-Kuhn-Tucker (KKT) residues of the sequence generated by augmented Lagrangian method (ALM) for solving convex composite conic programming (CCCP) has long been an outstanding open question. In this paper, we aim to resolve this issue by first conducting convergence rate analysis for the ALM with Rockafellar's stopping criteria under only a mild quadratic growth condition on the dual of CCCP. More importantly, by further assuming that the Robinson constraint qualification holds, we establish the R-superlinear convergence of the KKT residues of the iterative sequence under easy-to-implement stopping criteria {for} the augmented Lagrangian subproblems. Equipped with this discovery, we gain insightful interpretations on the impressive numerical performance of several recently developed semismooth Newton-CG based ALM solvers for solving linear and convex quadratic semidefinite programming.

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• Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Quadratic and Semi-Definite Programming
• QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming
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Cited by in corpus (5)

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• Kurdyka-Lojasiewicz Property of Zero-Norm Composite Functions
• Efficient sparse semismooth Newton methods for the clustered lasso problem
• On the Linear and Asymptotically Superlinear Convergence Rates of the Augmented Lagrangian Method with a Practical Relative Error Criterion
• Convergence of the Augmented Decomposition Algorithm