References
A. Shapiro, D. Dentcheva, and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia (2009).
A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization, Princeton Univ. Press, Princeton (2009).
R. T. Rockafellar, “Coherent approaches to risk in optimization under uncertainty,” in: Tutorials in Operations Research, INFORMS (2007), pp. 38–61.
K. Klamroth, E. Kobis, A. Schobel, and Chr. Tammer, “A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals,” Optimization, 62(5), 649–671 (2013).
Yu. M. Ermoliev, Stochastic Programming Methods [in Russian], Nauka, Moscow (1976).
P. Kouvelis and G. Yu, Robust Discrete Optimization and its Applications, Kluwer, Amsterdam (1997).
A. Ben-Tal and A. Nemirovski, “Robust solutions of linear programming problems contaminated with uncertain data,” Mathematical Programming, 88(3), 411–424 (2000).
M. Fischetty and M. Monaci, “Light robustness,” in: R. K. Ahuja, R. H. Mohring, and C. D. Zaroliagis (eds.), Robust and Online Large-Scale Optimization, Springer, Berlin (2009), pp. 61–84.
A. Ben-Tal, D. Bertsimas, and D. Brown, “A soft robust model for optimizing under ambiguity,” Operations Research, 58(4), 1220–1234 (2010).
A. Schobel, “Generalized light robustness and the trade-off between robustness and nominal quality,” Mathematical Methods of Operations Research, 80(2), 161–191 (2014).
P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, “Coherent measures of risk,” Mathematical Finance, 9(3), 203–228 (1999).
V. S. Kirilyuk, “Coherent risk measures and portfolio optimization problems,” Teoriya Optym. Rishen’, V. M. Glushkov Inst. of Cybernetics, NAS of Ukraine, Issue 2 (2003), pp. 111–119.
V. S. Kirilyuk, “The class of polyhedral coherent risk measures,” Cybern. Syst. Analysis, 40, No. 4, 599–690 (2004).
V. S. Kirilyuk, “Polyhedral coherent risk measures and optimal portfolios on the reward–risk ratio,” Cybern. Syst. Analysis, 50, No. 5, 724–740 (2014).
R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk," J. of Risk, 2(3), 21–41 (2000).
C. Acerbi, “Spectral measuers of risk: A coherent representation of subjective risk aversion,” J. of Banking and Finance, 26(7), 1505–1518 (2002).
S. Kusuoka, “On law invariant coherent risk measures,” in: S. Kusuoka and T. Maruyama (eds.), Advances in Mathematical Economics, 3, Springer, Tokyo (2001), pp. 83–95.
V. S. Kirilyuk, “Polyhedral coherent risk measures and investment portfolio optimization,” Cybern. Syst. Analysis, 44, No. 2, 250–260 (2008).
A. I. Kibzun, A. V. Naumov, and V. I. Norkin, “On reducing a quantile optimization problem with discrete distribution to a mixed integer programming problem,” Autom. and Remote Control, 74, No. 6, 951–967 (2013).
V. I. Norkin, A. I. Kibzun, and A. V. Naumov, “Reducing two-stage probabilistic optimization problems with discrete distribution of random data to mixed-integer programming problems,” Cybern. Syst. Analysis, 50, No. 5, 679–692 (2014).
R. T. Rockafellar, S. Uryasev, and M. Zabarankin, “Generalized deviations in risk analysis,” Finance and Stochastics, 10(1), 51–74 (2006).
V. S. Kirilyuk, “Expected utility theory, optimal portfolios, and polyhedral coherent risk measures,” Cybern. Syst. Analysis, 50, No. 6, 874–883 (2014).
S. Ahmed, “Convexity and decomposition of mean-risk stochastic programs,” Mathematical Programming, 106(3), 433–446 (2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2015, pp. 46–59.
Rights and permissions
About this article
Cite this article
Kirilyuk, V.S. Risk Measures in Stochastic Programming and Robust Optimization Problems. Cybern Syst Anal 51, 874–885 (2015). https://doi.org/10.1007/s10559-015-9780-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-015-9780-3