The Rate of Convergence of the Augmented Lagrangian Method for a Nonlinear Semidefinite Nuclear Norm Composite Optimization Problem

TL;DR

This paper analyzes the local convergence rate of the augmented Lagrangian method for nonlinear semidefinite nuclear norm optimization problems, establishing linear convergence under certain conditions without strict complementarity.

Contribution

It provides the first convergence rate analysis for this class of problems under nondegeneracy and second order conditions, using variational analysis techniques.

Findings

Convergence rate is linear and proportional to 1/c.

No strict complementarity required for convergence.

Analysis relies on variational properties of proximal mappings.

Abstract

We propose two basic assumptions, under which the rate of convergence of the augmented Lagrange method for a class of composite optimization problems is estimated. We analyze the rate of local convergence of the augmented Lagrangian method for a nonlinear semidefinite nuclear norm composite optimization problem by verifying these two basic assumptions. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold \bar c>0. The analysis is based on variational analysis about the proximal mapping of the nuclear norm and the projection operator onto the cone of positively semidefinite symmetric matrices.