Abstract
We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.
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References
R. Schultz, L. Stougie, M.H. van der Vlerk, Two-stage stochastic integer programming: A survey, Statist, Neerlandica 50 (1996) 404–416.
L. Stougie, Design and Analysis of Methods for Stochastic Integer Programming, CWI Tract 37, Amsterdam, 1987.
G. Laporte, F.V. Louveaux, The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett. 13 (1993) 133–142.
J. Tind, L.A. Wolsey, An elementary survey of general duality theory in mathematical programming, Math. Programming 21 (1981) 241–261.
L.A. Wolsey, A resource decomposition algorithm for general mathematical programs, Math. Programming Study 14 (1981) 244–257.
R. Schultz, On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Math. Programming 70 (1995) 73–89.
G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience, New York, 1988.
C.E. Blair, R.G. Jeroslow, The value function of an integer program, Math. Programming 23 (1982) 237–273.
R.M. Van Slyke, R.J.-B. Wets, L-shaped linear programs with application to optimal control and stochastic programming, SIAM J. Appl. Math. 17 (1969) 638–663.
L.A. Wolsey, Integer programming duality: Price functions and sensitivity analysis, Math. Programming 20 (1981) 173–195.
J. Tind, Multilevel optimization: A common framework for decomposition in general mathematical programming, Publication no. 93/2, Department of Operations Research, University of Aarhus, July 1993.
J.R. Birge, F.V. Louveaux, A multicut algorithm for two-stage stochastic linear programs, European J. Oper. Res. 34 (3) (1988) 384–392.
R.G. Jeroslow, Cutting plane theory: Algebraic methods, Discrete Math. 23 (1978) 121–150.
L. Schrage, L.A. Wolsey, Sensitivity analysis for branch and bound integer programming, Oper. Res. 33 (1985) 1008–1023.
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Carøe, C.C., Tind, J. L-shaped decomposition of two-stage stochastic programs with integer recourse. Mathematical Programming 83, 451–464 (1998). https://doi.org/10.1007/BF02680570
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DOI: https://doi.org/10.1007/BF02680570