Mathematical Programming

, Volume 108, Issue 2–3, pp 597–616 | Cite as

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables



In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.


Two-stage stochastic mixed-integer programs Benders' decomposition Convexification Reformulation-Linearization Technique (RLT) 

Mathematics Subject Classification (1991)

20E28 20G40 20C20 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Grado Department of Industrial and Systems Engineering (0118)Virginia Polytechnic Institute and State UniversityBlacksburgUSA

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