Abstract
Decomposition has proved to be one of the more effective tools for the solution of large-scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, which has been applied to both stochastic programming as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second-stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second-stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems that can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic mixed-integer programs.
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References
Ahmed, S., Tawarmalani, M., Sahinidis, N.V.: A finite branch and bound algorithm for two-stage stochastic integer programs. Math. Prog. 100, 355–377 (2004)
Balas, E.: Disjunctive programming. Annals of Discrete Mathematics 5, 3–51 (1979)
Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Prog. 58, 295–324 (1993)
Blair, C.: A closed-form representation of mixed-integer program value functions. Math. Prog. 71, 127–136 (1995)
Blair, C., Jeroslow, R.: The value function of an integer program. Math. Prog. 23, 237–273 (1982)
Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, Berlin, Germany, 1997
Caroe, C.C.: Decomposition in Stochastic Integer Programming. Ph.D. thesis, Institute of Mathematical Sciences, Dept. of Operations Research, University of Copenhagen, Denmark, 1998
Caroe, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Operations Research Letters 24, 37–45 (1999)
Jeroslow, R.: A cutting plane game for facial disjunctive programs. SIAM Journal on Control and Optimization 18, 264–281 (1980)
Klein Haneveld, W.K., van der Vlerk, M.H.: Stochastic integer programming: general models and algorithms. Annals of Operations Research 85, 39–57 (1999)
Laporte, G., Louveaux, F.V.: The integer L-shaped methods for stochastic integer programs with complete recourse. Operations Research Letters 13, 133–142 (1993)
Lovász, L., Schrijver, A.: Cones of matrices and set functions and 0-1 optimization. SIAM J. on Optimization 1, 166–190 (1991)
Lulli, G., Sen, S.: A branch and price algorithm for multi-stage stochastic integer programs with applications to stochastic lot sizing problems. Management Science 50, 786–796 (2004)
Ntaimo, L., Sen, S.: The million variable “march” for stochastic combinatorial optimization. To appear in Journal of Global Optimization 2003
Ruszczyński, A.: Decomposition methods in stochastic programming. Mathematical Programming B (Liebling and de Werra (eds.)), 79, 333–353 (1997)
Schultz, R.: On the structure and stability in stochastic programs with random technology matrix and complete integer recourse. Mathematical Programming 70, 73–89 (1995)
Schultz, R.: Stochastic programming with integer variables. Mathematical Programming Series B 97, 285–309 (2003)
Sen, S.: Decomposition algorithms for stochastic mixed-integer programming models. To appear in Handbook of Discrete Optimization. (K. Aardal, G. Nemhauser, R. Weismental, eds.), Elsevier Publishers, 2003
Sen, S., Higle, J.L.: The C 3 theorem and a D 2 algorithm for large scale stochastic optimization: set convexification. To appear in Mathematical Programming, 2004
Sen, S., Higle, J.L., Ntaimo, L.: A summary and illustration of disjunctive decomposition with set convexification. Stochastic Integer Programming and Network Interdiction Models. (D.L. Woodruff ed.), Kluwer Academic Press, Dordrecht, The Netherlands, 2002, pp. 105–125
Sen, S., Sherali, H.D.: A branch and bound algorithm for extreme point mathematical programming. Discrete Applied Mathematics 11, 265–280 (1985)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3, 411–430 (1990)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations and convex hull characterizations for mixed integer zero-one programming problems. Discrete Applied Mathematics 52, 83–106 (1994)
Sherali, H.D., Adams, W.P.: A Reformulation-Lizearization Technique for Solving Discrete and Continuous Nonconvex problems. Kluwer Academic Publishers, Boston, MA, 1999
Sherali, H.D., Fraticelli, B.M.P: 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)
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Sen, S., Sherali, H. Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math. Program. 106, 203–223 (2006). https://doi.org/10.1007/s10107-005-0592-5
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DOI: https://doi.org/10.1007/s10107-005-0592-5