The Augmented Lagrangian Methods: Overview and Recent Advances

Abstract. Large-scale constrained optimization is pivotal in modern scientific, engineering, and industrial computation, often involving complex systems with numerous variables and constraints. This paper provides a unified and comprehensive perspective on constructing augmented Lagrangian functions (based on the Hestenes–Powell–Rockafellar augmented Lagrangian) for various optimization problems, including nonlinear programming and convex and nonconvex composite programming. We present the augmented Lagrangian method (ALM), covering its theoretical foundations in both convex and nonconvex cases, and discuss several successful examples and applications. Recent advancements have extended the ALM’s capabilities to handle nonconvex constraints and ensure global convergence to first- and second-order stationary points. For nonsmooth convex problems, the ALM utilizes proximal operations, preserving desirable properties such as locally linear convergence rates. Furthermore, recent progress has refined the complexity analysis of the ALM and tackled challenging integer programming instances. This review aims to offer a thorough understanding of the ALM’s benefits and limitations, exploring different ALM variants designed to enhance convergence and computational performance. We also illustrate effective algorithms for ALM subproblems across different types of optimization problems and highlight practical implementations in several fields.