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.
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References
Adams, W.P. and H.D. Sherali (1993), Mixed-Integer Bilinear Programming Problems. Mathematical Programming 59(3), 279–305.
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.
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.
Birge, J.R. and M.A.H. Dempster (1966), Stochastic Programming Approaches to Stochastic Scheduling, Journal of Global Optimization 9(3–4), 417–451.
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.
Birge, J.R. and F.V. Louveaux (1997), Introduction to Stochastic Programming. New York, NY: Springer.
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.
Caroe, C.C. and R. Schultz (1999), Dual Decomposition in Stochastic Integer Programming. Operations Research Letters 24(1), 37–45.
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.
Caroe, C.C. and J. Tind (1998), L-Shaped Decomposition of two-Stage Stochastic Programs with Integer Recourse. Mathematical Programming 83(3), 451–464.
Geoffrion, A. and R. McBride (1978), Lagrangean Relaxation Applied to Capacitated Facility Location Problems. AIIE Transactions 10(1), 40–47.
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.
Higle, J.L. and S. Sen (1994), Conditional Stochastic Decomposition: An Algorithmic Interface for Optimization and Simulation. Operations Research 42(2), 311–322.
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.
Jeroslow, R. (1980), A Cutting Plane Game for Facial Disjunctive Programs. SIAM Journal on Control and Optimization 18(3), 264–280.
Kall, P. and S.W. Wallace (1994), Stochastic Programming. Chichester, England: John Wiley and Sons.
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.
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.
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.
Laporte, G., F.V. Louveaux, and H. Mercure (1992), The Vehicle Routing Problem with Stochastic Travel Times. Transportation Science 26(3), 161–170.
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.
Murphy, F.H., S. Sen, and A.L. Soyster (1982), Electric Utility Capacity Expansion Planning with Uncertain Load Forecasts. AIIE Transactions 14, 52–59.
Nemhauser, G.L. and L.A. Wolsey (1999), Integer and Combinatorial Optimization. New York, NY: Wiley-Interscience.
Ruszczynski, A. (1999), Some Advances in Decomposition Methods for Stochastic Linear Programming. Annals of Operations Research 85, 153–172.
Schultz, R. (1995), On Structure and Stability in Stochastic Programs with Random Technology Matrix and Complete Integer Recourse. Mathematical Programming 70(1), 73–90.
Schultz, R., L. Stougie, and M.H. van der Vlerk (1996), Two-Stage Stochastic Integer Programming: A Survey. Statistica Neerlandica 50(3), 404–416.
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.
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.
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.
Sherali, H.D. and W.P. Adams (1999), A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Boston, MA: Kluwer Academic Publishing.
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.
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.
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Sherali, H.D., Fraticelli, B.M. A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse. Journal of Global Optimization 22, 319–342 (2002). https://doi.org/10.1023/A:1013827731218
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DOI: https://doi.org/10.1023/A:1013827731218