Progressive Hedging Innovations for a Class of Stochastic Resource Allocation Problems

Progressive hedging (PH) is a scenario-based decomposition technique for solving stochastic programs. While PH has been successfully applied to a number of problems, a variety of issues arise when implementing PH in practice, especially when dealing with very difficult or large-scale mixed-integer problems. In particular, decisions must be made regarding the value of the penalty parameter p, criteria for termination, and techniques for accelerating convergence. We investigate these issues in the context of a class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce needs for sufficient combinations of resources. We introduce techniques for selecting p in proportion to the unit-cost of the associated decision variable, a mathematically-based heuristic for setting the p parameter, novel techniques for convergence acceleration, and methods for detecting and recovering from oscillatory behavior. The efficacy of these techniques is empirically assessed on two test cases: a difficult "laboratory" stochastic network flow problem and a largescale, real-world, robust spare parts procurement planning problem. Both test problems have integer variables in both stages. Additionally, we demonstrate that variable-specific p values are more effective than traditional fixed p values, and that the PH algorithm can serve as an effective heuristic even when the mathematical conditions for convergence are not respected.