Abstract
This paper is a contribution to the robustness analysis for stochastic programs whose set of feasible solutions depends on the probability distribution P. For various reasons, probability distribution P may not be precisely specified and we study robustness of results with respect to perturbations of P. The main tool is the contamination technique. For the optimal value, local contamination bounds are derived and applied to robustness analysis of the optimal value of a portfolio performance under risk-shaping CVaR constraints. A new robust portfolio efficiency test with respect to the second order stochastic dominance criterion is suggested and the contamination methodology is exploited to analyze its resistance with respect to additional scenarios.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonnans, J. S., & Shapiro, A. (2000). Perturbation analysis of optimization problems. Springer series in OR. New York.
Dentcheva, D., & Ruszczyński, A. (2010). Robust stochastic dominance and its application to risk-averse optimization. Mathematical Programming, 123, 85–100.
Dupačová, J. (1990). Stability and sensitivity analysis for stochastic programming. Annals of Operations Research, 27, 115–142.
Dupačová, J. (1996). Scenario-based stochastic programs: resistance with respect to sample. Annals of Operations Research, 64, 21–38.
Dupačová, J. (1998). Reflections on robust optimization. In K. Marti & P. Kall (Eds.), LNEMS: Vol. 458. Stochastic programming methods and technical applications (pp. 111–127). Berlin: Springer.
Dupačová, J. (2006). Stress testing via contamination. In K. Marti et al. (Ed.), LNEMS: Vol. 581. Coping with uncertainty, modeling and policy issues (pp. 29–46). Berlin: Springer.
Dupačová, J. (2008). Risk objectives in two-stage stochastic programming models. Kybernetika, 44, 227–242.
Dupačová, J., & Polívka, J. (2007). Stress testing for VaR and CVaR. Quantitative Finance, 7, 411–421.
Fiacco, A. (1983). Introduction to sensitivity and stability analysis in nonlinear programming. New York: Academic Press.
Gol’shtein, E. G. (1970). Vypukloje programmirovanije. Elementy teoriji. Moscow: Nauka. [Theory of convex programming, translations of mathematical monographs, Vol. 36, American Mathematical Society, Providence, 1972].
Kilianová, S., & Pflug, G. (2009). Optimal pension fund management under multi-period risk minimization. Annals of Operations Research, 166, 261–270.
Kopa, M. (2010). Measuring of second-order stochastic dominance portfolio efficiency. Kybernetika, 46, 488–500.
Kopa, M., & Chovanec, P. (2008). A second-order stochastic dominance portfolio efficiency measure. Kybernetika, 44, 243–258.
Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional-value-at-risk objective and constraints. The Journal of Risk, 2, 11–27.
Kuosmanen, T. (2004). Efficient diversification according to stochastic dominance criteria. Management Science, 50, 1390–1406.
Kyparisis, J., & Fiacco, A. (1987). Generalized convexity and concavity of the optimal value function in nonlinear programming. Mathematical Programming, 39, 285–304.
Ogryczak, W., & Ruszczyński, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13, 60–78.
Pagoncelli, B. K., Ahmed, S., & Shapiro, A. (2009). Sample average approximation method for chance constrained programming: theory and applications. Journal of Optimization Theory and Applications, 142, 399–416.
Pflug, G., & Wozabal, D. (2007). Ambiguity in portfolio selection. Quantitative Finance, 7, 435–442.
Prékopa, A. (1973). Contributions to the theory of stochastic programming. Mathematical Programming, 4, 202–221.
Prékopa, A. (2003), Probabilistic programming, Chap. 5 in Ruszczyński and Shapiro (2003) (pp. 267–351).
Robinson, S. M. (1987). Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity. Mathematical Programming Studies, 30, 45–66.
Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26, 1443–1471.
Römisch, W. (2003). Stability of stochastic programming problems. Chap. 8 in Ruszczyński and Shapiro (2003) (pp. 483–554).
Ruszczyński, A., & Shapiro, A. (Eds.) (2003). Stochastic programming. Handbooks in OR & MS, Vol. 10. Amsterdam: Elsevier.
Ruszczyński, A., & Vanderbei, R. J. (2003). Frontiers of stochastically nondominated portfolios. Econometrica, 71, 1287–1297.
Shapiro, A. (2003). Monte Carlo sampling methods, Chap. 6 in Ruszczyński and Shapiro (2003) (pp. 353–425).
Wang, W., & Ahmed, S. (2008). Sample average approximation for expected value constrained stochastic programs. OR Letters, 36, 515–519.
Wets, R. J.-B. (1989). Stochastic programming. In G. L. Nemhauser et al. (Ed.), Handbooks in OR & MS: Vol. 1. Optimization (pp. 573–629). Amsterdam: Elsevier. Chap. VIII.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dupačová, J., Kopa, M. Robustness in stochastic programs with risk constraints. Ann Oper Res 200, 55–74 (2012). https://doi.org/10.1007/s10479-010-0824-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-010-0824-9