A decomposition method for large scale MILPs, with performance guarantees and a power system application

Lagrangian duality in mixed integer optimization is a useful framework for problem decomposition and for producing tight lower bounds to the optimal objective. However, in contrast to the convex case, it is generally unable to produce optimal solutions directly. In fact, solutions recovered from the dual may not only be suboptimal, but even infeasible. In this paper we concentrate on large scale mixed-integer programs with a specific structure that appears in a variety of application domains such as power systems and supply chain management. We propose a solution method for these structures, in which the primal problem is modified in a certain way, guaranteeing that the solutions produced by the corresponding dual are feasible for the original unmodified primal problem. The modification is simple to implement and the method is amenable to distributed computation. We also demonstrate that the quality of the solutions recovered using our procedure improves as the problem size increases, making it particularly useful for large scale problem instances for which commercial solvers are inadequate. We illustrate the efficacy of our method with extensive experimentations on a problem stemming from power systems.