Surrogate "Level-Based" Lagrangian Relaxation for Mixed-Integer Linear Programming

Mixed-Integer Linear Programming (MILP) plays an important role across a range of scientific disciplines and within areas of strategic importance to society. The MILP problems, however, suffer from combinatorial complexity. Because of integer decision variables, as the problem size increases, the number of possible solutions increases super-linearly thereby leading to a drastic increase in the computational effort. To efficiently solve MILP problems, a "price-based" decomposition and coordination approach is developed to exploit 1. the super-linear reduction of complexity upon the decomposition and 2. the geometric convergence potential inherent to Polyak's step-sizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set step-sizes heuristically by adjusting hyper-parameters, the key novel way to obtain step-sizes is purely decision-based: a novel "auxiliary" constraint satisfaction problem is solved, from which the appropriate step-sizes are inferred. Testing results for large-scale Generalized Assignment Problems (GAP) demonstrate that for the majority of instances, certifiably optimal solutions are obtained. For stochastic job-shop scheduling as well as for pharmaceutical scheduling, computational results demonstrate the two orders of magnitude speedup as compared to frequently used Branch-and-Cut (B&C). SLBLR has a major impact on the efficient resolution of complex Mixed-Integer Programming (MIP) problems arising within a variety of scientific fields.