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：社会重要领域的运营问题常常被建模为混合整数线性规划（MILP）问题。尽管MILP问题遭受组合复杂性的困扰，Lagrangian松弛一直是通过分解解决相关难题的希望之光。由于Lagrangian对偶函数的非平滑特性，该方法的协调方面带来了严重的挑战。本文介绍了几个重大的历史里程碑（始于Polyak在1967年的先驱工作），以改善Lagrangian松弛通过改进非平滑泛函的优化来提高协调性。最后，本文介绍了用于快速解决MILP问题的Lagrangian松弛的最新发展。本文还简要讨论了此时此刻Lagrangian松弛可以提供的机会。