A study of progressive hedging for stochastic integer programming

Motivated by recent literature demonstrating the surprising effectiveness of the heuristic application of progressive hedging (PH) to stochastic mixed-integer programming (SMIP) problems, we provide theoretical support for the inclusion of integer variables, bridging the gap between theory and practice. We provide greater insight into the following observed phenomena of PH as applied to SMIP where optimal or at least feasible convergence is observed. We provide an analysis of a modified PH algorithm from a different viewpoint, drawing on the interleaving of (split) proximal-point methods (including PH), Gauss–Seidel methods, and the utilisation of variational analysis tools. Through this analysis, we show that under mild conditions, convergence to a feasible solution should be expected. In terms of convergence analysis, we provide two main contributions. First, we contribute insight into the convergence of proximal-point-like methods in the presence of integer variables via the introduction of the notion of persistent local minima. Secondly, we contribute an enhanced Gauss–Seidel convergence analysis that accommodates the variation of the objective function under mild assumptions. We provide a practical implementation of a modified PH and demonstrate its convergent behaviour with computational experiments in line with the provided analysis.