Surrogate “Level-Based” Lagrangian Relaxation for mixed-integer linear programming

• Published in: Scientific reports Vol. 12; no. 1; pp. 22417 - 12
• Main Authors: Bragin, Mikhail A., Tucker, Emily L.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 27.12.2022; Nature Publishing Group; Nature Portfolio
• Subjects: 639/166; 639/705; 692/700; Computer applications; Decomposition; Humanities and Social Sciences; Integer programming; Linear programming; multidisciplinary; Programming, Linear; Science; Science (multidisciplinary)
• ISSN: 2045-2322; 2045-2322
• DOI: 10.1038/s41598-022-26264-1

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 stepsizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set stepsizes heuristically by adjusting hyperparameters, the key novel way to obtain stepsizes is purely decision-based: a novel “auxiliary” constraint satisfaction problem is solved, from which the appropriate stepsizes are inferred. Testing results for large-scale Generalized Assignment Problems 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 Branch-and-Cut. The new method has a major impact on the efficient resolution of complex Mixed-Integer Programming problems arising within a variety of scientific fields.