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

arXiv: https://arxiv.org/abs/1706.08800 · HTML: https://mlsanity.com/paper/1706.08800/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2017.

## Built on by (most cited first, 3 of 3)

- [An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization](https://mlsanity.com/paper/2105.14033/) (2021)
- [An Inexact Halpern Iteration with Application to Distributionally Robust Optimization](https://mlsanity.com/paper/2402.06033/) (2024)
- [Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction](https://mlsanity.com/paper/2405.07965/) (2024)

## Abstract (arXiv)

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.