Inexact Augmented Lagrangian Methods for Conic Programs: Quadratic Growth and Linear Convergence

TL;DR

This paper proves that under strict complementarity, both primal and dual iterates of Augmented Lagrangian Methods for semidefinite programs converge linearly, resolving a longstanding open question about primal convergence rates.

Contribution

The paper establishes new quadratic growth and error bound properties for primal and dual SDPs, demonstrating linear convergence of primal iterates under mild assumptions.

Findings

Primal and dual iterates of ALMs converge linearly under strict complementarity.

New quadratic growth and error bound properties are proven for SDPs.

Addresses an open question about primal convergence in semidefinite optimization.

Abstract

Augmented Lagrangian Methods (ALMs) are widely employed in solving constrained optimizations, and some efficient solvers are developed based on this framework. Under the quadratic growth assumption, it is known that the dual iterates and the Karush-Kuhn-Tucker (KKT) residuals of ALMs applied to semidefinite programs (SDPs) converge linearly. In contrast, the convergence rate of the primal iterates has remained elusive. In this paper, we resolve this challenge by establishing new $\textit{quadratic growth}$ and $\textit{error bound}$ properties for primal and dual SDPs under the strict complementarity condition. Our main results reveal that both primal and dual iterates of the ALMs converge linearly contingent solely upon the assumption of strict complementarity and a bounded solution set. This finding provides a positive answer to an open question regarding the asymptotically linear convergence of the primal iterates of ALMs applied to semidefinite optimization.