Geodesic
Local stability and differentiability of the Mean–Conditional Value at Risk model defined on the mixed–integer loss functions
Kybernetika, Tome 46 (2010) no. 3, pp. 362-373

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In this paper, we study local stability of the mean-risk model with Conditional Value at Risk measure where the mixed-integer value function appears as a loss variable. This model has been recently introduced and studied in~Schulz and Tiedemann [16]. First, we generalize the qualitative results for the case with random technology matrix. We employ the contamination techniques to quantify a possible effect of changes in the underlying probability distribution on the optimal value. We use the generalized qualitative results to express the explicit formula for the directional derivative of the local optimal value function with respect to the underlying probability measure. The derivative is used to construct the bounds. Similarly, we can approximate the behavior of the local optimal value function with respect to the changes of the risk-aversion parameter which determines our aversion to risk.
In this paper, we study local stability of the mean-risk model with Conditional Value at Risk measure where the mixed-integer value function appears as a loss variable. This model has been recently introduced and studied in~Schulz and Tiedemann [16]. First, we generalize the qualitative results for the case with random technology matrix. We employ the contamination techniques to quantify a possible effect of changes in the underlying probability distribution on the optimal value. We use the generalized qualitative results to express the explicit formula for the directional derivative of the local optimal value function with respect to the underlying probability measure. The derivative is used to construct the bounds. Similarly, we can approximate the behavior of the local optimal value function with respect to the changes of the risk-aversion parameter which determines our aversion to risk.
Classification : 90C11, 90C15, 90C31, 91B28, 91B30
Keywords: mean-CVaR model; mixed-integer value function; stability analysis; contamination techniques; derivatives of optimal value function 
Branda, Martin. Local stability and differentiability of the Mean–Conditional Value at Risk model defined on the mixed–integer loss functions. Kybernetika, Tome 46 (2010) no. 3, pp. 362-373. http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a1/
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     title = {Local stability and differentiability of the {Mean{\textendash}Conditional} {Value} at {Risk} model defined on the mixed{\textendash}integer loss functions},
     journal = {Kybernetika},
     pages = {362--373},
     year = {2010},
     volume = {46},
     number = {3},
     mrnumber = {2676075},
     zbl = {1202.90203},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a1/}
}
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[1] Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Akademie-Verlag, Berlin 1982. | Zbl

[2] Billingsley, P.: Convergence of Probability Measures. (Wiley Series in Probability and Statistics.) Second edition. Wiley, New York 1999. | MR | Zbl

[3] Blair, C. E., Jeroslow, R. G.: The value function of a mixed integer program: I. Discrete Mathematics 19 (1977), 121–138. | DOI | MR | Zbl

[4] Dobiáš, P.: Contamination for stochastic integer programs. Bulletin of the Czech Econometric Society 10 (2003), No. 18.

[5] Dupačová, J.: Stability in stochastic programming with recourse. Contaminated distributions. Mathematical Programming Study 27 (1986), 133–144. | DOI | MR

[6] Dupačová, J.: Stability and sensitivity-analysis for stochastic programming. Ann. Oper. Res. 27 (1990), 115–142. | DOI | MR

[7] Dupačová, J.: Output analysis for approximated stochastic programs. In: Stochastic Optimization: Algorithms and Applications (S. Uryasev and P. M. Pardalos, Eds.), Kluwer Academic Publishers, Dordrecht 2001, pp. 1–29. | MR

[8] Dupačová, J.: Risk objectives in two-stage stochastic programming models. Kybernetika 44 (2008), 2, 227–242. | MR

[9] Dupačová, J., Polívka, J.: Stress testing for VaR and CVaR. Quantitative Finance 27 (2007), 4, 411–421. | DOI | MR

[10] Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Dordrecht 1999. | MR | Zbl

[11] Rockafellar, T. R., Uryasev, S.: Conditional value-at-risk for genera loss distributions. J. Banking and Finance 26 (2002), 1443–1471. | DOI

[12] Römisch, W.: Stability of Stochastic Programming Problems. In: Stochastic Programming (A. Ruszczynski and A. Shapiro eds.), Handbooks in Operations Research and Management Science Vol. 10, Elsevier, Amsterdam (2003), 483-554. | MR

[13] Römisch, W., Schultz, R.: Multistage stochastic integer programming: an introduction. In: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, and J. Rambau, eds.), Springer-Verlag, Berlin 2001, pp. 581–600.

[14] Schultz, R.: On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Mathematical Programming 26 (1995), 73–89. | DOI | MR | Zbl

[15] Schultz, R.: Stochastic programming with integer variables. Mathematical Programming, Ser. B 97 (2003), 285–309. | MR | Zbl

[16] Schultz, R., Tiedemann, S.: Conditional value-at-risk in stochastic programs with mixed-integer recourse. Mathematical Programming, Ser. B 105 (2006), 365–386. | DOI | MR | Zbl

[17] Szegö, G.: Risk Measures for the 21st Century. Wiley, Chichester 2004.

[18] Wallace, S. W., Ziemba, W. T.: Applications of Stochastic Programming. (MPS-SIAM Book Series on Optimization, Volume 5.) SIAM Philadelpia 2005. | Zbl