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On the Convex Hull of the Composition of a Separable and a Linear Function

Author

Listed:
  • KLEIN HANEVELD, Willem K.

    (Department of Econometrics, University of Groningen)

  • STOUGIE, Leen

    (Institute for Actuarial Sciences, Econometrics and Operations Research, University)

  • VAN der VLERK, M.H.

    (CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium)

Abstract

Theorems on the convex hull of an extended real-valued function living on a Hilbert space are presented in case the function is separable and in case the function is a composition of another function and a linear transformation. Equivalence of convex hulls of functions and their biconjugates is used. In particular, it is shown that under properness conditions the biconjugate of a separable function is equal to the sum of the biconjugates of its constituents, and the biconjugate of a composition of a function and a bounded linear transformation is equal to the composition of the biconjugate of the function and the linear transformation on the range of the linear transformation. The results are applied to describe the convex hull of the objective function of a problem in stochastic integer programming.

Suggested Citation

  • 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).
    • Handle: RePEc:cor:louvco:1995070
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1995.html
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    Citations

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    Cited by:

    1. repec:dgr:rugsom:03a01 is not listed on IDEAS
    2. Klein Haneveld, W.K. & Vlerk, M.H. van der, 2000. "Optimizing electricity distribution using two-stage integer recourse models," Research Report 00A26, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    3. Vlerk, Maarten H. van der, 2002. "On multiple simple recourse models," Research Report 02A06, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    4. repec:dgr:rugsom:02a21 is not listed on IDEAS
    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. Vlerk, Maarten H. van der, 2002. "Convex approximations for complete integer recourse models," Research Report 02A21, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    7. 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).
    8. 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).

    More about this item

    Keywords

    convex hull; conjugate function; simple integer recourse;
    All these keywords.

    JEL classification:

    • B25 - Schools of Economic Thought and Methodology - - History of Economic Thought since 1925 - - - Historical; Institutional; Evolutionary; Austrian; Stockholm School
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General

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