A Sampling-based Progressive Hedging Algorithm for Stochastic Programming

The progressive hedging algorithm (PHA) is a cornerstone among algorithms for large-scale stochastic programming problems. However, its traditional implementation is hindered by some limitations, including the requirement to solve all scenario subproblems in each iteration, reliance on an explicit probability distribution, and a convergence process that is highly sensitive to the choice of certain penalty parameters. This paper introduces a sampling-based PHA which aims to overcome these limitations. Our approach employs a dynamic selection process for the number of scenario subproblems solved per iteration. It incorporates adaptive sequential sampling for determining sample sizes, a stochastic conjugate subgradient method for direction finding, and a line-search technique to update the dual variables. Experimental results demonstrate that this novel algorithm not only addresses the bottlenecks of the conventional PHA but also potentially surpasses its scalability, representing a substantial improvement in the field of stochastic programming.