  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.
