On the Asymptotic Superlinear Convergence of the Augmented Lagrangian Method for Semidefinite Programming with Multiple Solutions

TL;DR

This paper analyzes the asymptotic superlinear convergence of the augmented Lagrangian method for convex semidefinite programming, especially when the primal problem has multiple solutions, extending previous results that required uniqueness.

Contribution

It provides the first superlinear convergence analysis of ALM for SDP with multiple solutions under mild growth conditions, broadening understanding of solver efficiency.

Findings

Primal sequence converges superlinearly under mild conditions.

Dual feasibility and objective converge superlinearly.

Conditions for dual convergence are established.

Abstract

Solving large scale convex semidefinite programming (SDP) problems has long been a challenging task numerically. Fortunately, several powerful solvers including SDPNAL, SDPNAL+ and QSDPNAL have recently been developed to solve linear and convex quadratic SDP problems to high accuracy successfully. These solvers are based on the augmented Lagrangian method (ALM) applied to the dual problems with the subproblems being solved by semismooth Newton-CG methods. Noticeably, thanks to Rockafellar's general theory on the proximal point algorithms, the primal iteration sequence generated by the ALM enjoys an asymptotic Q-superlinear convergence rate under a second order sufficient condition {for the primal problem}. This second order sufficient condition implies that the primal problem has a unique solution, which can be restrictive in many applications. For gaining more insightful interpretations on the high efficiency of these solvers, in this paper we conduct an asymptotic superlinear convergence analysis of the ALM for convex SDP when the primal problem has multiple solutions (can be unbounded). Under a fairly mild second order growth condition, we prove that the primal iteration sequence generated by the ALM converges asymptotically Q-superlinearly, while the dual feasibility and the dual objective function value converge asymptotically R-superlinearly. Moreover, by studying the metric subregularity of the Karush-Kuhn-Tucker solution mapping, we also provide sufficient conditions to guarantee the asymptotic R-superlinear convergence of the dual iterate.