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Simple Integer Recourse Models: Convexity and Convex Approximations

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  • Klein Haneveld, Willem K.
  • Stougie, Leen
  • Vlerk, Maarten H. van der

    (Groningen University)

Abstract

We consider the objective function of a simple recourse problem with fixed technology matrix and integer second-stage variables. Separability due to the simple recourse structure allows to study a one-dimensional version instead. Based on an explicit formula for the objective function, we derive a complete description of the class of probability density functions such that the objective function is convex. This result is also stated in terms of random variables. Next, we present a class of convex approximations of the objective function, which are obtained by perturbing the distributions of the right-hand side parameters. We derive a uniform bound on the absolute error of the approximation. Finally, we give a representation of convex simple integer recourse problems as continuous simple recourse problems, so that they can be solved by existing special purpose algorithms

Suggested Citation

  • Klein Haneveld, Willem K. & Stougie, Leen & Vlerk, Maarten H. van der, 2004. "Simple Integer Recourse Models: Convexity and Convex Approximations," Research Report 04A21, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    • Handle: RePEc:gro:rugsom:04a21
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    File URL: http://irs.ub.rug.nl/ppn/288251970
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    References listed on IDEAS

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    1. 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," LIDAM Reprints CORE 1215, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Willem Klein Haneveld & Maarten van der Vlerk, 1999. "Stochastic integer programming:General models and algorithms," Annals of Operations Research, Springer, vol. 85(0), pages 39-57, January.
    3. repec:dgr:rugsom:03a01 is not listed on IDEAS
    4. Rüdiger Schultz, 1993. "Continuity Properties of Expectation Functions in Stochastic Integer Programming," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 578-589, August.
    5. Vlerk, Maarten H. van der, 2003. "Simplification of recourse models by modification of recourse data," Research Report 03A01, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    6. KLEIN HANEVELD, Willem K. & STOUGIE, Leen & VAN der VLERK, M.H., 1995. "On the Convex Hull of the Composition of a Separable and a Linear Function," LIDAM Discussion Papers CORE 1995070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Vlerk, Maarten H. van der, 2004. "Convex approximations for a class of mixed-integer recourse models," Research Report 04A28, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
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