Advertisement

Cybernetics and Systems Analysis

, Volume 55, Issue 6, pp 999–1008 | Cite as

Polyhedral Coherent Risk Measures and Robust Optimization

  • V. S. KirilyukEmail author
Article
  • 10 Downloads

Abstract

Properties of the apparatus of polyhedral coherent risk measures, its relationship with problems of robust and distributionally robust optimization, as well as its application under uncertainty are described. Problems of calculating robust structures of polyhedral coherent risk measures and their minimization, which are reduced to the corresponding linear programming problems, are considered.

Keywords

polyhedral coherent risk measure Conditional Value-at-Risk robust optimization distributionally robust optimization uncertainty set linear programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. S. Kirilyuk, “The class of polyhedral coherent risk measures,” Cybern. Syst. Analysis, Vol. 40, No. 4, 599–609 (2004).MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. S. Kirilyuk, “Polyhedral coherent risk measures and investment portfolio optimization,” Cybern. Syst. Analysis, Vol. 44, No. 2, 250–260 (2008).MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. S. Kirilyuk, “Polyhedral coherent risk measures and optimal portfolios on the reward–risk ratio,” Cybern. Syst. Analysis, Vol. 50, No. 5, 724–740 (2014).MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. S. Kirilyuk, “Expected utility theory, optimal portfolios, and polyhedral coherent risk measures,” Cybern. Syst. Analysis, Vol. 50, No. 6, 874–883 (2014).MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. S. Kirilyuk, “Polyhedral coherent risk measures in the case of imprecise scenario estimates,” Cybern. Syst. Analysis, Vol. 54, No. 3, 423–433 (2018).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. S. Kirilyuk, “Risk measures in stochastic programming and robust optimization problems,” Cybern. Syst. Analysis, Vol. 51, No. 6, 874–885 (2015).MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. Delage and Y. Ye, “Distributionally robust optimization under moment uncertainty with application to data-driven problems,” Operations Research, Vol. 58, No. 3, 595–612 (2010).MathSciNetCrossRefGoogle Scholar
  8. 8.
    W. Wiesemann, D. Kuhn, and M. Sim, “Distributionally robust convex optimization,” Operations Research, Vol. 62, No. 6, 1358–1376 (2014).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Shapiro, “Distributionally robust stochastic programming,” SIAM J. Optim., Vol. 27, No. 4, 2258–2275 (2017).MathSciNetCrossRefGoogle Scholar
  10. 10.
    T. Augustin, F. Coolen, G. Cooman, and M. Troffaes (eds.), Introduction to Imprecise Probabilities, Wiley, Chichester (2014).zbMATHGoogle Scholar
  11. 11.
    P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, “Coherent measures of risk,” Mathematical Finance, Vol. 9, No. 3, 203–228 (1999).MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. T. Rockafellar and S. Uryasev, “Optimization of Conditional Value-at-Risk,” J. of Risk, Vol. 2, No. 3, 21–41 (2000).CrossRefGoogle Scholar
  13. 13.
    C. Acerbi, “Spectral measures of risk: A coherent representation of subjective risk aversion,” J. Banking & Finance, Vol. 26, No. 7, 1505–1518 (2002).Google Scholar
  14. 14.
    S. Kusuoka, “On law invariant coherent risk measures,” in: S. Kusuoka and T. Maruyama (eds.), Advances in Mathematical Economics, Vol. 3, Springer, Tokyo (2001), pp. 83–95.CrossRefGoogle Scholar
  15. 15.
    D. Bertsimas and D. B. Brown, “Constructing uncertainty sets for robust linear optimization,” Operations Research, Vol. 57, No. 6, 1483–1495 (2009).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations