We present a method for computing lower bounds in the Progressive Hedging Algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. We show that the best possible lower bound obtained using dual prices is as tight as the lower bound obtained using the Dual Decomposition method. We report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds. \keywords{Stochastic mixed-integer programming \and Decomposition algorithms \an
Citation
Technical Report; Graduate School of Management; UC Davis; Davis CA 95616; October 11, 2013; revised January 21 2014 and May 2014