Solving two-stage stochastic programming problems using alternating direction method of multipliers

This study introduces a proximal three-block Alternating Direction Method of Multipliers (ADMM) variant with a projected dual update for solving two-stage stochastic programming problems. We first recast the two-stage stochastic program into a deterministic three-block separable optimization problem, a formulation designed to leverage the parallel computation capabilities inherent in ADMM. A key contribution is the establishment of theoretical convergence for our proposed regularized ADMM variant when applied to this specific three-block formulation; this analysis utilizes principles of lower semi-continuity and the Kurdyka-Lojasiewicz (K-L) property. The parallel implementation of this approach significantly accelerates the solution process, which is crucial for effectively handling large-scale stochastic programming problems. Through numerical analysis on a suite of benchmark problems, including the large-scale storm instance and a non-convex example, this study validates the robust performance of the proposed ADMM approach. Comparisons with Progressive Hedging, a proximal ADMM variant, and direct Gurobi solution of the extensive form highlight its efficiency in terms of computational time and solution quality, particularly for larger problems. This research, therefore, contributes a theoretically sound and practically effective ADMM framework for two-stage stochastic programming, offering valuable insights for broader optimization contexts.