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

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.