Ordinal-Optimization Concept Enabled Decomposition and Coordination of Mixed-Integer Linear Programming Problems

Many important optimization problems, such as manufacturing scheduling and power system unit commitment, are formulated as Mixed-Integer Linear Programming (MILP) problems. Such problems are generally difficult to solve because of their combinatorial nature, and may subject to strict computation time limitations. Recently, our decomposition-and-coordination method “Surrogate Absolute Value Lagrangian Relaxation” (SAVLR) exploits the exponential reduction of complexity upon problem decomposition and effectively coordinates subproblem solutions. In the method, subproblems are generally solved by using Branch-and-Cut (B&C). When subproblems are complicated, however, the approach might not be able to generate high-quality solutions within time limitations. In this paper, motivated by the “Ordinal Optimization” concept, this difficulty is resolved through exploiting a specific property of SAVLR that subproblem solutions only need to be “good enough” to satisfy a convergence condition. Time consuming B&C is eliminated in many iterations through obtaining “good enough” subproblem solutions based on “crude models” (e.g., LP-relaxed problems) or from heuristics. Testing results on generalized assignment problems demonstrate that the approach obtains high-quality solutions in a computationally efficient manner and significantly outperforms other approaches. This approach also opens up a new way to solve practical MILP problems that are subject to strict computation time limitations.