Abstract
We consider mixed-integer recourse (MIR) models with a single recourse constraint. We relate the second-stage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alonso-Ayuso, A., Escudero, L. F., & Ortuño, M. T. (2003). BFC, a branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0-1 programs. European Journal of Operational Research, 151(3), 503–519.
Birge, J. R., & Louveaux, F. V. (1997). Introduction to stochastic programming. New York: Springer.
Carøe, C. C., & Tind, J. (1997). A cutting-plane approach to mixed 0-1 stochastic integer programs. European Journal of Operational Research, 101(2), 306–316.
Dyer, M., & Stougie, L. (2006). Computational complexity of stochastic programming problems. Mathematical Programming, 106(3, Ser. A), 423–432.
Guan, Y., Ahmed, S., & Nemhauser, G. L. (2005). Sequential pairing of mixed integer inequalities. In Lecture notes in comput. sci. : Vol. 3509. Integer programming and combinatorial optimization (pp. 23–34). Berlin: Springer.
Kall, P., & Wallace, S. W. (1994). Stochastic programming. New York: Wiley. Also available as PDF file e.g. via http://stoprog.org.
Klein Haneveld, W. K., & van der Vlerk, M. H. (1999). Stochastic integer programming: general models and algorithms. Annals of Operation Research, 85, 39–57.
Klein Haneveld, W. K., Stougie, L., & van der Vlerk, M. H. (1995). On the convex hull of the simple integer recourse objective function. Annals of Operation Research, 56, 209–224.
Klein Haneveld, W. K., Stougie, L., & van der Vlerk, M. H. (1996). An algorithm for the construction of convex hulls in simple integer recourse programming. Annals of Operation Research, 64, 67–81.
Klein Haneveld, W. K., Stougie, L., & van der Vlerk, M. H. (2006). Simple integer recourse models: convexity and convex approximations. Mathematical Programming, 108(2–3), 435–473.
Kong, N., Schaefer, A. J., & Hunsaker, B. (2006). Two-stage integer programs with stochastic right-hand sides: a superadditive dual approach. Mathematical Programming, 108(2–3), 275–296.
Laporte, G., & Louveaux, F. V. (1993). The integer L-shaped method for stochastic integer programs with complete recourse. Operations Research Letters, 13, 133–142.
Louveaux, F. V., & Schultz, R. (2003). Stochastic integer programming. In A. Ruszczynski & A. Shapiro (Eds.), Handbooks in operations research and management science : Vol. 10. Stochastic programming (Chap. 4, pp. 213–266). Amsterdam: Elsevier.
Louveaux, F. V., & van der Vlerk, M. H. (1993). Stochastic programming with simple integer recourse. Mathematical Programming, 61, 301–325.
Mayer, J. (1998). Optimization theory and applications : Vol. 1. Stochastic linear programming algorithms: a comparison based on a model management system. New York: Gordon and Breach.
Norkin, V. I., Ermoliev, Yu. M., & Ruszczyński, A. (1998). On optimal allocation of indivisibles under uncertainty. Operational Research, 46(3), 381–395.
Prékopa, A. (1995). Stochastic programming. Dordrecht: Kluwer Academic.
Ruszczynski, A., & Shapiro, A. (2003). Handbooks in operations research and management science : Vol. 10. Stochastic programming. Amsterdam: Elsevier.
Schultz, R. (1993). Continuity properties of expectation functions in stochastic integer programming. Mathematics of Operations Research, 18, 578–589.
Schultz, R. (1995). On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Mathematical Programming, 70, 73–89.
Schultz, R., & Tiedemann, S. (2003). Risk aversion via excess probabilities in stochastic programs with mixed-integer recourse. SIAM Journal of Optimization (electronic), 14(1), 115–138.
Schultz, R., & Tiedemann, S. (2006). Conditional value-at-risk in stochastic programs with mixed-integer recourse. Mathematical Programming, 105(2–3), 365–386.
Sen, S., & Higle, J. L. (2005). The C3 theorem and a D2 algorithm for large scale stochastic mixed-integer programming: set convexification. Mathematical Programming, 104(1), 1–20.
Sen, S., & Sherali, H. D. (2006). Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Mathematical Programming, 106(2), 203–223.
Sherali, H. D., & Zhu, X. (2006). On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables. Mathematical Programming, 108(2–3), 597–616.
Silva, F., & Wood, R. K. (2006). Solving a class of stochastic mixed-integer programs with branch and price. Mathematical Programming, 108(2–3), 395–418.
Stougie, L., & van der Vlerk, M. H. (1997). Stochastic integer programming. In M. Dell’Amico & F. Maffioli (Eds.), Annotated bibliographies in combinatorial optimization (Chap. 9, pp. 127–141). New York: Wiley.
Stougie, L., & van der Vlerk, M. H. (2003). Approximation in stochastic integer programming. Research Report 2003–12, Stochastic Programming E-Print Series, http://www.speps.org.
Swamy, C., & Shmoys, D. B. (2005). Sampling-based approximation algorithms for multi-stage stochastic optimization. In 46th annual IEEE symposium on foundations of computer science (FOCS’05) (pp. 357–366)
van der Vlerk, M. H. (1995). Stochastic programming with integer recourse. Ph.D. thesis, University of Groningen, The Netherlands.
van der Vlerk, M. H. (2003). Simplification of recourse models by modification of recourse data. In K. Marti & Y. Ermoliev (Eds.), Lecture notes in economics and mathematical systems : Vol. 532. Dynamic stochastic optimization (pp. 321–336). Berlin: Springer.
van der Vlerk, M. H. (2004). Convex approximations for complete integer recourse models. Mathematical Programming, 99(2), 297–310.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Van der Vlerk, M.H. Convex approximations for a class of mixed-integer recourse models. Ann Oper Res 177, 139–150 (2010). https://doi.org/10.1007/s10479-009-0591-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-009-0591-7