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Journal of Global Optimization

, Volume 22, Issue 1–4, pp 319–342 | Cite as

A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse

  • Hanif D. SheraliEmail author
  • Barbara M.P. Fraticelli
Article

Abstract

In this paper, we modify Benders' decomposition method by using concepts from the Reformulation-Linearization Technique (RLT) and lift-and-project cuts in order to develop an approch for solving discrete optimization problems that yield integral subproblems, such as those that arise in the case of two-stage stochastic programs with integer recourse. We first demonstrate that if a particular convex hull representation of the problem's constrained region is available when binariness is enforced on only the second-stage (or recourse) variables, then the regular Benders' algorithm is applicable. The proposed procedure is based on sequentially generating a suitable partial description of this convex hull representation as needed in the process of deriving valid Benders' cuts. The key idea is to solve the subproblems using an RLT or lift-and-project cutting plane scheme, but to generate and store the cuts as functions of the first-stage variables. Hence, we are able to re-use these cutting planes from one subproblem solution to the next simply by updating the values of the first-stage decisions. The proposed Benders' cuts also recognize these RLT or lift-and-project cuts as functions of the first-stage variables, and are hence shown to be globally valid, thereby leading to an overall finitely convergent solution procedure. Some illustrative examples are provided to elucidate the proposed approach. The focus of this paper is on developing such a finitely convergent Benders' approach for problems having 0-1 mixed-integer subproblems as in the aforementioned context of two-stage stochastic programs with integer recourse. A second part of this paper will deal with related computational experiments.

Keywords

Hull Stochastic Program Discrete Optimization Decomposition Algorithm Plane Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Adams, W.P. and H.D. Sherali (1993), Mixed-Integer Bilinear Programming Problems. Mathematical Programming 59(3), 279–305.Google Scholar
  2. Ahmed, S., M. Tawarmalani, and N.V. Sahindis (2000), A Finite Branch and Bound Algorithm for Two-Stage Stochastic Integer Programs. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA.Google Scholar
  3. Balas, E., S. Ceria and G. Cornuejols (1993), A Lift-and-Project Cutting Plane Algorithms for Mixed 0–1 Programs. Mathematical Programming 58, 295–324.Google Scholar
  4. Birge, J.R. and M.A.H. Dempster (1966), Stochastic Programming Approaches to Stochastic Scheduling, Journal of Global Optimization 9(3–4), 417–451.Google Scholar
  5. Birge, J.R. and F.V. Louveaux (1988), A Multicut Algorithm for Two-Stage Stochastic Linear Programs. European Journal of Operational Research 34(3), 384–392.Google Scholar
  6. Birge, J.R. and F.V. Louveaux (1997), Introduction to Stochastic Programming. New York, NY: Springer.Google Scholar
  7. Carino, D.R., T. Kent, D.H. Meyers, C. Stacy, M. Sylvanus, A.L. Turner, K. Watanabe, and W.T. Ziemba (1994), The Russell-Yasuda Kasai Model: An Asset / Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming. Interfaces 24(1), 29–49.Google Scholar
  8. Caroe, C.C. and R. Schultz (1999), Dual Decomposition in Stochastic Integer Programming. Operations Research Letters 24(1), 37–45.Google Scholar
  9. Caroe, C.C. and J. Tind (1997), A Cutting-Plane Approach to Mixed 0–1 Stochastic Integer Programs. European Journal of Operational Research 101(2), 306–316.Google Scholar
  10. Caroe, C.C. and J. Tind (1998), L-Shaped Decomposition of two-Stage Stochastic Programs with Integer Recourse. Mathematical Programming 83(3), 451–464.Google Scholar
  11. Geoffrion, A. and R. McBride (1978), Lagrangean Relaxation Applied to Capacitated Facility Location Problems. AIIE Transactions 10(1), 40–47.Google Scholar
  12. Higle, J.L. and S. Sen (1991), Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse. Mathematics of Operations Research 16(3), 650–669.Google Scholar
  13. Higle, J.L. and S. Sen (1994), Conditional Stochastic Decomposition: An Algorithmic Interface for Optimization and Simulation. Operations Research 42(2), 311–322.Google Scholar
  14. Higle, J.L. and S. Sen (2000), The C Theorem and a D Algorithm for Large Scale Stochastic Integer Programming: Set Convexification. Working paper, Department of Systems and Industrial Engineering, The University of Arizona, Tucson, AZ 85721. Also presented at the 17th International Symposium on Mathematical Programming, Atlanta, GA, August 7–11, 2000.Google Scholar
  15. Jeroslow, R. (1980), A Cutting Plane Game for Facial Disjunctive Programs. SIAM Journal on Control and Optimization 18(3), 264–280.Google Scholar
  16. Kall, P. and S.W. Wallace (1994), Stochastic Programming. Chichester, England: John Wiley and Sons.Google Scholar
  17. Klein Haneveld, W.K., L. Stougie, and M.H. van der Vlerk (1996), An Algorithm for the Construction of Convex Hulls in Simple Integer Recourse Programming. Annals of Operations Research 64, 67–81.Google Scholar
  18. Klein Haneveld, W.K. and M.H. van der Vlerk (1999), Stochastic Integer Programming: General Models and Algorithms. Annals of Operations Research 85, 39–57.Google Scholar
  19. Laporte, G. and F.V. Louveaux (1993), The Integer L-Shaped Method for Stochastic Integer Programs with Complete Recourse. Operations Research Letters 13(3), 133–142.Google Scholar
  20. Laporte, G., F.V. Louveaux, and H. Mercure (1992), The Vehicle Routing Problem with Stochastic Travel Times. Transportation Science 26(3), 161–170.Google Scholar
  21. Laporte, G., F.V. Louveaux, and L. van Hamme (1994), Exact Solution of a Stochastic Location Problem b an Integer L-Shaped Algorithm. Transportation Science 28(2), 95–103.Google Scholar
  22. Murphy, F.H., S. Sen, and A.L. Soyster (1982), Electric Utility Capacity Expansion Planning with Uncertain Load Forecasts. AIIE Transactions 14, 52–59.Google Scholar
  23. Nemhauser, G.L. and L.A. Wolsey (1999), Integer and Combinatorial Optimization. New York, NY: Wiley-Interscience.Google Scholar
  24. Ruszczynski, A. (1999), Some Advances in Decomposition Methods for Stochastic Linear Programming. Annals of Operations Research 85, 153–172.Google Scholar
  25. Schultz, R. (1995), On Structure and Stability in Stochastic Programs with Random Technology Matrix and Complete Integer Recourse. Mathematical Programming 70(1), 73–90.Google Scholar
  26. Schultz, R., L. Stougie, and M.H. van der Vlerk (1996), Two-Stage Stochastic Integer Programming: A Survey. Statistica Neerlandica 50(3), 404–416.Google Scholar
  27. Schultz, R., L. Stougie, and M.H. van der Vlerk (1998), Solving Stochastic Programs with Integer Recourse by Enumeration: A Framework using Grobner Basis Reductions. Mathematical Programming 83(2), 229–252.Google Scholar
  28. Sherali, H.D. and W.P. Adams (1990), A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems. SIAM Journal on Discrete Mathematics 3(3), 411–430.Google Scholar
  29. Sherali, H.D. and W.P. Adams (1994), A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-Integer Zero-One Programming Problems. Discrete Applied Mathematics 52(1), 83–106.Google Scholar
  30. Sherali, H.D. and W.P. Adams (1999), A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Boston, MA: Kluwer Academic Publishing.Google Scholar
  31. Sherali, H.D., Y. Lee, and Y. Kim (2000), Partial Convexification Cuts. Manuscript, Grado Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. Also presented at the 17th International Symposium on Mathematical Programming, Atlanta, GA, August 7–11, 2000.Google Scholar
  32. Van Slyke, R.M. and R. Wets (1969), L-Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming. SIAM Journal of Applied Mathematics 17(4), 638–663.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

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

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