Abstract
In this paper we analyze robust approaches to decision making under uncertainty where the expected outcome is maximized but the probabilities are known imprecisely. A conservative robust approach takes into account any probability distribution thus leading to the notion of robustness focusing on the worst case scenario and resulting in the max-min optimization. We consider softer robust models allowing the probabilities to vary only within given intervals. We show that the robust solution for only upper bounded probabilities becomes the tail mean, known also as the conditional value-at-risk (CVaR), with an appropriate tolerance level. For proportional upper and lower probability limits the corresponding robust solution may be expressed by the optimization of appropriately combined the mean and the tail mean criteria. Finally, a general robust solution for any arbitrary intervals of probabilities can be expressed with the optimization problem very similar to the tail mean and thereby easily implementable with auxiliary linear inequalities.
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References
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent masures of risk. Mathematical Finance, 9, 203–228.
Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009). Robust Optimization. Princeton: Princeton University Press.
Bertsimas, D., & Thiele, A. (2006). Robust and data-driven optimization: modern decision making under uncertainty. Tutorials on Operations Research, INFORMS, Chap. 4, 195–122.
Dupacova, J. (1987). Stochastic programming with incomplete information: A survey of results on postoptimization and sensitivity analysis. Optimization, 18, 507–532.
Embrechts, P., Klüppelberg, C., & Mikosch T. (1997). Modelling Extremal Events for Insurance and Finance. New York: Springer.
Ermoliev, Y., Gaivoronski, A., & Nedeva, C. (1985). Stochastic optimization problems with incomplete information on distribution functions. SIAM Journal on Control and Optimization, 23, 697–716.
Ermoliev, Y., & Hordijk, L. (2006). Facets of robust decisions. In K. Marti, Y. Ermoliev, M. Makowski, & G. Pflug (Eds.), Coping with Uncertainty, Modeling and Policy Issues (pp. 4–28). Berlin: Springer.
Ermoliev, Y., & Leonardi, G. (1982). Some proposals for stochastic facility location models. Mathematical Modelling, 3, 407–420.
Ermoliev, Y., & Wets, R.J.-B. (1988). Stochastic programming, an introduction. Numerical techniques for stochastic optimization. Springer Series in Computational Mathematics, 10, 1–32.
Haimes, Y. Y. (1993). Risk of extreme events and the fallacy of the expected value. Control and Cybernetics, 22, 7–31.
Hampel, F., Ronchetti, E., Rousseeuw, P., & Stahel, W. (1986). Robust Statistics: The Approach Based on Influence Function. New York: Wiley.
Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35, 73–101.
Kostreva, M. M., Ogryczak, W., & Wierzbicki, A. (2004). Equitable aggregations and multiple criteria analysis. European Journal of Operational Research, 158, 362–367.
Kouvelis, P., & Yu, G. (1997). Robust Discrete Optimization and Its Applications. Dordrecht: Kluwer.
Krzemienowski, A., & Ogryczak, W. (2005). On extending the LP computable risk measures to account downside risk. Computational Optimization and Applications, 32, 133–160.
Lim, C., Sherali, H. D., & Uryasev, S. (2010). Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Computational Optimization and Applications, 46, 391–415.
Mansini, R., Ogryczak, W., Speranza, M. G. (2003). On LP solvable models for portfolio selection. Informatica, 14, 37–62.
Mansini, R., Ogryczak, W., & Speranza, M. G. (2007). Conditional value at risk and related linear programming models for portfolio optimization. Annals of Operations Research, 152, 227–256.
Miettinen, K., Deb, K., Jahn, J., Ogryczak, W., Shimoyama, K., & Vetchera, R. (2008). Future challenges (Chap. 16). In Multi-Objective Optimization – Evolutionary and Interactive Approaches, Lecture Notes in Computer Science, vol. 5252 (pp. 435–461). New York: Springer.
Miller, N., & Ruszczyński, A. (2008). Risk-adjusted probability measures in portfolio optimization with coherent measures of risk. European Journal of Operational Research, 191, 193–206.
Ogryczak, W. (1999). Stochastic dominance relation and linear risk measures. In A. M. J. Skulimowski (Ed.), Financial Modelling – Proc. 23rd Meeting EURO WG Financial Modelling, Cracow, 1998 (pp. 191–212). Cracow: Progress & Business Publisher.
Ogryczak, W. (2000). Multiple criteria linear programming model for portfolio selection. Annals of Operations Research, 97, 143–162.
Ogryczak, W. (2002). Multiple criteria optimization and decisions under risk. Control and Cybernetics, 31, 975–1003.
Ogryczak, W., & Ruszczyński, A. (2001). On consistency of stochastic dominance and mean-semideviation models. Mathematical Programming, 89, 217–232.
Ogryczak, W., & Ruszczyński, A. (2002). Dual stochastic dominance and quantile risk measures. International Transactions on Operational Research, 9, 661–680.
Ogryczak, W., & Śliwiński, T. (2010). On solving the dual for portfolio selection by optimizing conditional value at risk. Computational Optimization and Applications, DOI: 10.1007/s10589-010-9321-y.
Perny, P., Spanjaard, O., & Storme, L.-X. (2006). A decision-theoretic approach to robust optimization in multivalued graphs. Annals of Operations Research, 147, 317–341.
Pflug, G.Ch. (2001). Scenario tree generation for multiperiod financial optimization by optimal discretization. Mathematical Programming, 89, 251–271.
Pflug, G.Ch. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In S. Uryasev (Ed.), Probabilistic Constrained Optimization: Methodology and Applications. Dordrecht: Kluwer.
Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior and Organization, 3, 323–343.
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41.
Roell, A. (1987). Risk aversion in Quiggin and Yaari’s rank-order model of choice under uncertainty. Economic Journal, 97, 143–159.
Ruszczyński, A., & Shapiro, A. (Eds.) (2003). Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10 (pp. 272–281). Elsevier
Takeda, A., & Kanamori, T. (2009). A robust approach based on conditional value-at-risk measure to statistical learning problems. European Journal of Operational Research, 198, 287–296.
Thiele, A. (2008). Robust stochastic programming with uncertain probabilities. IMA Journal of Management Mathematics, 19, 289–321.
Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica 55, 95–115.
Acknowledgements
The research was partially supported by the Polish National Budget Funds 2009–2011 for science under the grant N N516 3757 36.
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Ogryczak, W. (2012). Robust Decisions under Risk for Imprecise Probabilities. In: Ermoliev, Y., Makowski, M., Marti, K. (eds) Managing Safety of Heterogeneous Systems. Lecture Notes in Economics and Mathematical Systems, vol 658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22884-1_3
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DOI: https://doi.org/10.1007/978-3-642-22884-1_3
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