A Decomposition Method for Large Scale MILPs, with Performance Guarantees and a Power System Application

TL;DR

This paper introduces a decomposition method for large-scale mixed-integer linear programs that guarantees feasible solutions and improves solution quality with increasing problem size, especially useful in power system applications.

Contribution

It proposes a simple, implementable modification to primal problems that ensures dual solutions are feasible and effective for large-scale structured MILPs, with demonstrated power system application.

Findings

Solution quality improves as problem size increases.

Method guarantees feasible solutions from duals.

Effective in power system optimization problems.

Abstract

Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce optimal solutions directly. In fact, solutions recovered from the dual may be not only suboptimal, but even infeasible. In this paper we concentrate on large scale mixed--integer programs with a specific structure that is of practical interest, as it appears in a variety of application domains such as power systems or 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 computations. 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 instances for which commercial solvers are inadequate. We illustrate the efficacy of our method with extensive experimentations on a problem stemming from power systems.