• Michael Zabarankin
  • Stan Uryasev
Part of the Springer Optimization and Its Applications book series (SOIA, volume 85)


This chapter discusses two classification methods: logistic regression and support vector machines (SVMs). Both methods are popular in various applications ranging from biomedicine and bioinformatics to image recognition and credit scoring. The logistic regression can classify a training data into several categories, whereas SVMs are mostly binary classifiers, i.e., deal with two classification categories.


  1. [1]
    Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Financ. 26(7), 1487–1503 (2002)CrossRefGoogle Scholar
  2. [2]
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    Bonami, P., Lejeune, M.A.: An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Oper. Res. 57(3), 650–670 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    Buckley, J.J.: Entropy principles in decision making under risk. Risk Anal. 5(4), 303–313 (1979)CrossRefMathSciNetGoogle Scholar
  5. [5]
    Chang, C.C., Lin, C.J.: Training ν-support vector classifiers: theory and algorithms. Neural Comput. 13, 2119–2147 (2001)CrossRefMATHGoogle Scholar
  6. [6]
    Chekhlov, A., Uryasev, S., Zabarankin, M.: Portfolio Optimization with Drawdown Constraints, pp. 263–278. Risk Books, London (2003)Google Scholar
  7. [7]
    Chekhlov, A., Uryasev, S., Zabarankin, M.: Drawdown measure in portfolio optimization. Int. J. Theor. Appl. Financ. 8(1), 13–58 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Cortes, C., Vapnik, V.: Support vector networks. Mach. Learn. 20, 273–297 (1995)MATHGoogle Scholar
  9. [9]
    Costa, J., Hero, A., Vignat, C.: On Solutions to Multivariate Maximum-entropy Problems, vol. 2683, pp. 211–228. Springer, Berlin (2003)Google Scholar
  10. [10]
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, New York (2006)MATHGoogle Scholar
  11. [11]
    Cozzolino, J.M., Zahner, M.J.: The maximum-entropy distribution of the future market price of a stock. Oper. Res. 21(6), 1200–1211 (1973)CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    Crisp, D.J., Burges, C.J.C.: A geometric interpretation of ν-SVM classifiers. Neural Inf. Process. Syst. 12, 244–250 (2000)Google Scholar
  13. [13]
    Fölmer, H., Schied, A.: Stochastic Finance, 2nd edn. Walter de Gruyter GmbH & Co., Berlin (2004)CrossRefGoogle Scholar
  14. [14]
    Grauer, R.R.: Introduction to asset pricing theory and tests. In: Roll, R. (ed.) The International Library of Critical Writings in Financial Economics. Edward Elgar Publishing Inc., Cheltenham (2001)Google Scholar
  15. [15]
    Grechuk, B., Zabarankin, M.: Inverse portfolio problem with mean-deviation model. Eur. J. Oper. Res. (2013, to appear)Google Scholar
  16. [16]
    Grechuk, B., Molyboha, A., Zabarankin, M.: Maximum entropy principle with general deviation measures. Math. Oper. Res. 34(2), 445–467 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    Grechuk, B., Molyboha, A., Zabarankin, M.: Chebyshev’s inequalities with law invariant deviation measures. Probab. Eng. Informational Sci. 24, 145–170 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Hardy, G.E., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, New York (1952)MATHGoogle Scholar
  19. [19]
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edn. Springer, New York (2008)Google Scholar
  20. [20]
    Hull, J.C., White, A.D.: Valuing credit derivatives using an implied copula approach. J. Derivatives 14(2), 8–28 (2006)CrossRefGoogle Scholar
  21. [21]
    Iscoe, I., Kreinin, A., Mausser, H., Romanko, A.: Portfolio credit-risk optimization. J. Bank. Financ. 36(6), 1604–1615 (2012)CrossRefGoogle Scholar
  22. [22]
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    Jensen, J.L.: Surles fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30(1), 175–193 (1906)CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    Johnson, O., Vignat, C.: Some results concerning maximum Rényi entropy distributions. Annales de l’Institut Henri Poincare (B) Probab. Stat. 43(3), 339–351 (2007)Google Scholar
  25. [25]
    Kalinchenko, K., Uryasev, S., Rockafellar, R.T.: Calibrating risk preferences with generalized CAPM based on mixed CVaR deviation. J. Risk 15(1), 45–70 (2012)Google Scholar
  26. [26]
    Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33–50 (1978)CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    Kurdila, A., Zabarankin, M.: Convex Functional Analysis. Birkhauser, Switzerland (2005)MATHGoogle Scholar
  28. [28]
    Levy, H.: Stochastic dominance and expected utility: survey and analysis. Manag. Sci. 38(4), 555–593 (1992)CrossRefMATHGoogle Scholar
  29. [29]
    Markowitz, H.M.: Portfolio selection. J. Financ. 7(1), 77–91 (1952)Google Scholar
  30. [30]
    Markowitz, H.M.: Foundations of portfolio theory. J. Financ. 46, 469–477 (1991)CrossRefGoogle Scholar
  31. [31]
    Mercer, J.: Functions of positive and negative type, and their connection with the theory of integral equations. Philos. Trans. Roy. Soc. London 209(441–458), 415–446 (1909)CrossRefMATHGoogle Scholar
  32. [32]
    Molyboha, A., Zabarankin, M.: Stochastic optimization of sensor placement for diver detection. Oper. Res. 60(2), 292–312 (2012)CrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    Ogryczak, W., Ruszczyński, A.: On consistency of stochastic dominance and mean-semideviation models. Math. Program. 89, 217–232 (2001)CrossRefMATHMathSciNetGoogle Scholar
  34. [34]
    Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13(1), 60–78 (2002)CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    Perez-Cruz, F., Weston, J., Hermann, D.J.L., Schölkopf, B.: Extension of the ν-SVM range for classification. Adv. Learn. Theory Method. Models Appl. 190, 179–196 (2003)Google Scholar
  36. [36]
    Rockafellar, R.T.: Convex Analysis, Princeton Mathematics Series, vol. 28. Princeton University Press, Princeton (1970)Google Scholar
  37. [37]
    Rockafellar, R.T.: Coherent approaches to risk in optimization under uncertainty. In: Gray, P. (ed.) Tutorials in Operations Research, pp. 38–61. INFORMS, Hanover (2007)Google Scholar
  38. [38]
    Rockafellar, R.T., Royset, J.O.: On buffered failure probability in design and optimization of structures. Reliab. Eng. Syst. Saf. 95, 499–510 (2011)CrossRefGoogle Scholar
  39. [39]
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)Google Scholar
  40. [40]
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Financ. 26(7), 1443–1471 (2002)CrossRefGoogle Scholar
  41. [41]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Deviation measures in risk analysis and optimization. Technical Report 2002–7. ISE Department, University of Florida, Gainesville, FL (2002)Google Scholar
  42. [42]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Generalized deviations in risk analysis. Financ. Stoch. 10(1), 51–74 (2006)CrossRefMATHMathSciNetGoogle Scholar
  43. [43]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Master funds in portfolio analysis with general deviation measures. J. Bank. Financ. 30(2), 743–778 (2006)CrossRefMathSciNetGoogle Scholar
  44. [44]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Optimality conditions in portfolio analysis with general deviation measures. Math. Program. 108(2–3), 515–540 (2006)CrossRefMATHMathSciNetGoogle Scholar
  45. [45]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Equilibrium with investors using a diversity of deviation measures. J. Bank. Financ. 31(11), 3251–3268 (2007)CrossRefGoogle Scholar
  46. [46]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Risk tuning with generalized linear regression. Math. Oper. Res. 33(3), 712–729 (2008)CrossRefMATHMathSciNetGoogle Scholar
  47. [47]
    Roell, A.: Risk aversion in Quiggin and Yaari’s rank-order model of choice under uncertainty. Econ. J. 97(Issue Supplement: Conference papers), 143–159 (1987)Google Scholar
  48. [48]
    Rousseeuw, P.J., Driessen, K.: Computing LTS regression for large data sets. Data Min. Knowl. Discov. 12(1), 29–45 (2006)CrossRefMathSciNetGoogle Scholar
  49. [49]
    Rousseeuw, P., Leroy, A.: Robust Regression and Outlier Detection. Wiley, New York (1987)CrossRefMATHGoogle Scholar
  50. [50]
    Roy, A.D.: Safety first and the holding of assets. Econometrica 20(3), 431–449 (1952)CrossRefMATHGoogle Scholar
  51. [51]
    Ruszczyński, A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)MATHGoogle Scholar
  52. [52]
    Schölkopf, B., Smola, A., Williamson, R., Bartlett, P.: New support vector algorithms. Neural Comput. 12, 1207–1245 (2000)CrossRefGoogle Scholar
  53. [53]
    Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Financ. 19, 425–442 (1964)Google Scholar
  54. [54]
    Sharpe, W.F.: Capital asset prices with and without negative holdings. J. Financ. 46, 489–509 (1991)CrossRefGoogle Scholar
  55. [55]
    Takeda, A., Sugiyama, M.: ν-support vector machine as conditional value-at-risk minimization. In: Proceedings of the 25th International Conference on Machine Learning (ICML 2008), pp. 1056–1063. Morgan Kaufmann, Montreal, Canada (2008)Google Scholar
  56. [56]
    Thomas, M.U.: A generalized maximum entropy principle. Oper. Res. 27(6), 1188–1196 (1979)CrossRefMATHGoogle Scholar
  57. [57]
    Tobin, J.: Liquidity preference as behavior towards risk. Rev. Econ. Stud. 25(2), 65–86 (1958)CrossRefGoogle Scholar
  58. [58]
    Tsyurmasto, P., Zabarankin, M., Uryasev, S.: Value-at-risk support vector machine: stability to outliers. J. Comb. Optim. (2014, to appear)Google Scholar
  59. [59]
    Venables, W., Ripley, B.: Modern Applied Statistics with S-PLUS, 4th edn. Springer, New York (2002)Google Scholar
  60. [60]
    van der Waerden, B.: Mathematische Statistik. Springer, Berlin (1957)MATHGoogle Scholar
  61. [61]
    Wets, R.J.B.: Statistical estimation from an optimization viewpoint. Ann. Oper. Res. 85, 79–101 (1999)CrossRefMATHMathSciNetGoogle Scholar
  62. [62]
    Yaari, M.E.: The dual theory of choice under risk. Econometrica 55(1), 95–115 (1987)CrossRefMATHMathSciNetGoogle Scholar
  63. [63]
    Zabarankin, M., Pavlikov, K., Uryasev, S.: Capital asset pricing model (CAPM) with drawdown measure. Eur. J. Oper. Res. (2013, to appear)Google Scholar

Copyright information

© Springer Science+Business Media, New York 2014

Authors and Affiliations

  • Michael Zabarankin
    • 1
  • Stan Uryasev
    • 2
  1. 1.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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