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.
Bibliographical noteFunding Information:
The authors would like to thank the associate editor and two anonymous referees for their helpful comments on improving the quality of this paper. Thanks also go to Professor Terry Rockafellar for his comments on the unboundedness of the Lagrangian multipliers of the KKT solution mapping of the original form leading to the current form of Example 2 during his visit to the Hong Kong Polytechnic University.
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- Augmented Lagrangian method
- Convex composite conic programming
- Implementable criteria
- Quadratic growth condition