Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

We present a method for computing lower bounds in the Progressive Hedging Algorithm (PHA) for two-stage and multi-stage stochastic mixedinteger 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. This research was sponsored in part by the US Department of Energy’s ARPA-e Green Energy Network Integration (GENI) program, and by the Department of Energy’s Office of Science, Advanced Scientific Computing Research program. Thanks to Ge Guo for assistance with the numerical results. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94-AL85000. Dinakar Gade Sabre Holdings Southlake, TX 76092 Jean-Paul Watson Sandia National Laboratories, Albuquerque, NM 87185 Gabriel Hackebeil Texas A&M University, College Station, TX 77843 Sarah M. Ryan Iowa State University, Ames, IA, 50011 Roger J-B Wets, David L. Woodruff University of California Davis, Davis, CA 95616 2 Dinakar Gade et al.