Operations in areas of importance to society are frequently modeled as Mixed-Integer Linear Programming (MILP) problems. While MILP problems suffer from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve the associated difficulties through decomposition. Due to the non-smooth nature of Lagrangian dual functions, the coordination aspect of the method has posed serious challenges. This paper presents several significant historical milestones (beginning with Polyak's pioneering work in 1967) toward improving Lagrangian Relaxation coordination through improved optimization of non-smooth functionals. Finally, this paper presents the most recent developments in Lagrangian Relaxation for fast resolution of MILP problems. The paper also briefly discusses the opportunities that Lagrangian Relaxation can provide at this point in time.

翻译：对社会重要领域的运作常被建模为混合整数线性规划（MILP）问题。尽管MILP问题受制于组合复杂性，但拉格朗日松弛法通过分解技术为解决相关难题带来了希望。由于拉格朗日对偶函数具有非光滑特性，该方法的协调环节始终面临严峻挑战。本文梳理了提升拉格朗日松弛协调性能的若干重要历史里程碑（始于1967年Polyak的开创性工作），其核心是通过改进非光滑函数优化实现突破。最后，本文阐述了拉格朗日松弛法在快速求解MILP问题方面的最新进展，并简要探讨了当前该技术所蕴含的发展机遇。