Parallel algorithms for two-stage stochastic optimization

We develop scalable algorithms for two-stage stochastic program optimizations. We propose performance optimizations such as cut-window mechanism in Stage 1 and scenario clustering in Stage 2 of benders method for solving two-stage stochastic programs. A naive implementation of benders method has slow convergence rate and does not scale well to large number of processors especially when the problem size is large and/or there are integer variables in Stage 1. Parallelization of stochastic integer programs pose very unique characteristics that make them very challenging to parallelize. We develop a Parallel Stochastic Integer Program Solver (PSIPS) that exploits nested parallelism by exploring the branch-and-bound tree vertices in parallel along with scenario parallelization. PSIPS has been shown to have high parallel efficiency of greater than 40% at 120 cores which is significantly greater than the parallel efficiency of state-of-the-art mixed-integer program solvers. A significant portion of the time in this branch-and-bound solver is spent in optimizing the stochastic linear program at the root vertex. Stochastic linear programs at other vertices of the branch-and-bound tree take very less iterations to converge because they can inherit benders cut from their parent vertices and/or the root. Therefore, it is important to reduce the optimization time of the stochastic linear program at the root vertex. We propose two decomposition schemes namely the Split-and-Merge (SAM) method and the Lagrangian Decomposition and Merge (LDAM) method that significantly increase the convergence rate of benders decomposition. SAM method gives up to 64% reduction in solution time while also giving significantly higher parallel speedups as compared to the naive benders method. LDAM method, on the other hand, has made it possible to solve otherwise intractable stochastic programs. We further provide a computational engine for many real-time and dynamic problems faced by US Air Mobility Command. We first propose a stochastic programming solution to the military aircraft allocation problem with consideration for disaster management. Then, we study US AMC's dynamic mission re-planning problem and propose a mathematical formulation that is computationally feasible and leads to significant savings in cost as compared to myopic and deterministic optimization. It is expected that this work will provide the springboard for more robust problem solving with HPC in many logistics and planning problems