Fenchel Decomposition for Stochastic Mixed-IntegerProgramming

Abstract

This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD usesa class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. We derive FD cuts based on both the first and second stage variables, and devise an FD algorithm for SMIP with binary first stage and establish finite convergence for mixed-binary second stage. We also derive alternative FD cuts based on the second stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first stage variables. We then devise an FD-L algorithm based on the lifted FD cuts. Finally, we report on preliminary computational results based on example instances from the literature. The results are promising and show the lifted FD cuts to have better performance than the regular FD cuts. Furthermore, both the FD and FD-L algorithms outperform a standard solver on large-scaleinstances.