An augmented Lagrangian method for strongly regular minimizers in a class of convex composite optimization problems

TL;DR

This paper develops an augmented Lagrangian method for convex composite optimization problems, establishing theoretical equivalences and guarantees, and demonstrates its effectiveness through numerical experiments on entropy optimization.

Contribution

It introduces a novel augmented Lagrangian approach with theoretical analysis for strongly regular minimizers in convex composite problems.

Findings

Theoretical equivalence between second-order conditions and constraint qualifications.

Established nonsingularity conditions for ALM subproblems.

Numerical validation on von Neumann entropy optimization.

Abstract

In this paper, we study a class of convex composite optimization problems. We begin by characterizing the equivalence between the primal/dual strong second-order sufficient condition and the dual/primal nondegeneracy condition. Building on this foundation, we derive a specific set of equivalent conditions for the perturbation analysis of the problem. Furthermore, we employ the augmented Lagrangian method (ALM) to solve the problem and provide theoretical guarantees for its performance. Specifically, we establish the equivalence between the primal/dual second-order sufficient condition and the dual/primal strict Robinson constraint qualification, as well as the equivalence between the dual nondegeneracy condition and the nonsingularity of Clarke's generalized Jacobian for the ALM subproblem. These theoretical results form a solid foundation for designing efficient algorithms. Finally, we apply the ALM to the von Neumann entropy optimization problem and present numerical experiments to demonstrate the algorithm's effectiveness.