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A Note on Stability for Risk-Averse Stochastic Complementarity Problems

Author

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  • Johanna Burtscheidt

    (University of Duisburg-Essen)

  • Matthias Claus

    (University of Duisburg-Essen)

Abstract

In this paper, we propose a new approach to stochastic complementarity problems which allows to take into account various notions of risk aversion. Our model enhances the expected residual minimization formulation of Chen and Fukushima (Math Oper Res 30:1022–1038, 2005) by replacing the expectation with a more general convex, nondecreasing and law-invariant risk functional. Relevant examples of such risk functionals include the Expected Excess and the Conditional Value-at-Risk. We examine qualitative stability of the resulting one-stage stochastic optimization problem with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the decision vector and the entering probability measure. By a classical result from parametric optimization, this implies upper semicontinuity of the optimal value function. Throughout the analysis, we assume for building the model that a nonlinear complementarity function fulfills a certain polynomial growth condition. We conclude the paper demonstrating that this assumption holds in the vast majority of all practically relevant cases.

Suggested Citation

  • Johanna Burtscheidt & Matthias Claus, 2017. "A Note on Stability for Risk-Averse Stochastic Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 298-308, January.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:1:d:10.1007_s10957-016-1020-0
    DOI: 10.1007/s10957-016-1020-0
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    References listed on IDEAS

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    1. Xiaojun Chen & Masao Fukushima, 2005. "Expected Residual Minimization Method for Stochastic Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 1022-1038, November.
    2. Aurél Galántai, 2012. "Properties and construction of NCP functions," Computational Optimization and Applications, Springer, vol. 52(3), pages 805-824, July.
    3. Rüdiger Schultz, 2000. "Some Aspects of Stability in Stochastic Programming," Annals of Operations Research, Springer, vol. 100(1), pages 55-84, December.
    4. Kenji Hamatani & Masao Fukushima, 2011. "Pricing American options with uncertain volatility through stochastic linear complementarity models," Computational Optimization and Applications, Springer, vol. 50(2), pages 263-286, October.
    5. Zhang, Chao & Chen, Xiaojun & Sumalee, Agachai, 2011. "Robust Wardrop's user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach," Transportation Research Part B: Methodological, Elsevier, vol. 45(3), pages 534-552, March.
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