Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms

• 成果类型: Article; Early Access
• 署名作者: Andrews, Roland; Carpentier, Justin; Taylor, Adrien
• 署名单位: Inria; Universite PSL; Ecole Normale Superieure (ENS); Communaute Universite Grenoble Alpes; Centre National de la Recherche Scientifique (CNRS); Universite Grenoble Alpes (UGA); Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02418-1
• 发表日期: 2026-09-14
• 关键词: 1st-order methods; performance

摘要

This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization problems. We demonstrate that, under mild assumptions, the sequences of iterates generated by ALMs converge to solutions of the closest feasible problem. We establish progressively stronger convergence results, ranging from basic sequence convergence to more precise convergence rates, under a hierarchy of assumptions. This study leverages the classical relationship between ALMs and the proximal-point algorithm applied to the dual problem. A key technical contribution is a set of concise results on the behavior of the proximal-point algorithm when applied to functions that may lack minimizers. These results pertain to its convergence in terms of its subgradients and of the values of the convex conjugate.