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Stochastic linear programming with a distortion risk constraint

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  • Bazovkin, Pavel
  • Mosler, Karl

Abstract

Linear optimization problems are investigated whose parameters are uncertain. We apply coherent distortion risk measures to capture the violation of restrictions. Such a model turns out to be appropriate for many applications and, principally, for the mean-risk portfolio selection problem. Each risk constraint induces an uncertainty set of coefficients, which comes out to be a weighted-mean trimmed region. We consider a problem with a single constraint. Given an external sample of the coefficients, the uncertainty set is a convex polytope that can be exactly calculated. If the sample is i.i.d. from a general probability distribution, the solution of the stochastic linear program (SLP) is a consistent estimator of the SLP solution with respect to the underlying probability. An efficient geometrical algorithm is proposed to solve the SLP.

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  • Bazovkin, Pavel & Mosler, Karl, 2011. "Stochastic linear programming with a distortion risk constraint," Discussion Papers in Econometrics and Statistics 6/11, University of Cologne, Institute of Econometrics and Statistics.
    • Handle: RePEc:zbw:ucdpse:611
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    References listed on IDEAS

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    1. Wenqing Chen & Melvyn Sim & Jie Sun & Chung-Piaw Teo, 2010. "From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization," Operations Research, INFORMS, vol. 58(2), pages 470-485, April.
    2. Bazovkin, Pavel & Mosler, Karl, 2012. "An Exact Algorithm for Weighted-Mean Trimmed Regions in Any Dimension," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 47(i13).
    3. Pflug Georg Ch., 2006. "On distortion functionals," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 1-16, July.
    4. Pflug Georg Ch., 2006. "On distortion functionals," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 45-60, July.
    5. Dimitris Bertsimas & David B. Brown, 2009. "Constructing Uncertainty Sets for Robust Linear Optimization," Operations Research, INFORMS, vol. 57(6), pages 1483-1495, December.
    6. Karthik Natarajan & Dessislava Pachamanova & Melvyn Sim, 2009. "Constructing Risk Measures from Uncertainty Sets," Operations Research, INFORMS, vol. 57(5), pages 1129-1141, October.
    7. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    8. Dyckerhoff, Rainer & Mosler, Karl, 2012. "Weighted-mean regions of a probability distribution," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 318-325.
    9. Frank Fabozzi & Dashan Huang & Guofu Zhou, 2010. "Robust portfolios: contributions from operations research and finance," Annals of Operations Research, Springer, vol. 176(1), pages 191-220, April.
    10. Adam, Alexandre & Houkari, Mohamed & Laurent, Jean-Paul, 2008. "Spectral risk measures and portfolio selection," Journal of Banking & Finance, Elsevier, vol. 32(9), pages 1870-1882, September.
    11. Dyckerhoff, Rainer & Mosler, Karl, 2011. "Weighted-mean trimming of multivariate data," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 405-421, March.
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    Cited by:

    1. Postek, K.S. & den Hertog, D. & Melenberg, B., 2014. "Tractable Counterparts of Distributionally Robust Constraints on Risk Measures," Discussion Paper 2014-031, Tilburg University, Center for Economic Research.
    2. Postek, K.S. & den Hertog, D. & Melenberg, B., 2015. "Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures (revision of CentER DP 2014-031)," Discussion Paper 2015-047, Tilburg University, Center for Economic Research.
    3. Bazovkin, Pavel, 2014. "Geometrical framework for robust portfolio optimization," Discussion Papers in Econometrics and Statistics 01/14, University of Cologne, Institute of Econometrics and Statistics.

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