Computational Optimization and Applications

, Volume 46, Issue 3, pp 391–415 | Cite as

Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization

  • Churlzu LimEmail author
  • Hanif D. Sherali
  • Stan Uryasev


Conditional Value-at-Risk (CVaR) is a portfolio evaluation function having appealing features such as sub-additivity and convexity. Although the CVaR function is nondifferentiable, scenario-based CVaR minimization problems can be reformulated as linear programs (LPs) that afford solutions via widely-used commercial softwares. However, finding solutions through LP formulations for problems having many financial instruments and a large number of price scenarios can be time-consuming as the dimension of the problem greatly increases. In this paper, we propose a two-phase approach that is suitable for solving CVaR minimization problems having a large number of price scenarios. In the first phase, conventional differentiable optimization techniques are used while circumventing nondifferentiable points, and in the second phase, we employ a theoretically convergent, variable target value nondifferentiable optimization technique. The resultant two-phase procedure guarantees infinite convergence to optimality. As an optional third phase, we additionally perform a switchover to a simplex solver starting with a crash basis obtained from the second phase when finite convergence to an exact optimum is desired. This three phase procedure substantially reduces the effort required in comparison with the direct use of a commercial stand-alone simplex solver (CPLEX 9.0). Moreover, the two-phase method provides highly-accurate near-optimal solutions with a significantly improved performance over the interior point barrier implementation of CPLEX 9.0 as well, especially when the number of scenarios is large. We also provide some benchmarking results on using an alternative popular proximal bundle nondifferentiable optimization technique.


Portfolio optimization CVaR Nondifferentiable optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersson, F., Mausser, H., Rosen, D., Uryasev, S.: Credit risk optimization with conditional value-at-risk criterion. Math. Program. 89(2), 273–291 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barahona, F., Anbil, R.: The volume algorithm: Producing primal solutions with a subgradient method. Math. Program. 87(3), 385–399 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barr, D.R., Slezak, N.L.: A comparison of multivariate normal generators. Commun. ACM 15(12), 1048–1049 (1972) zbMATHCrossRefGoogle Scholar
  5. 5.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006) zbMATHGoogle Scholar
  6. 6.
    Bitran, G., Hax, A.: On the solution of convex knapsack problems with bounded variables. In: Proceedings of IXth International Symposium on Mathematical Programming, Budapest, pp. 357–367 (1976) Google Scholar
  7. 7.
    Brännlund, U.: On relaxation methods for nonsmooth convex optimization. Ph.D. Dissertation, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden (1993) Google Scholar
  8. 8.
    Butenko, S., Golodnikov, A., Uryasev, S.: Optimal security liquidation algorithms. Comput. Optim. Appl. 32(1), 9–27 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goffin, J.L., Kiwiel, K.C.: Convergence of a simple subgradient level method. Math. Program. 85(1), 207–211 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Held, M., Wolfe, P., Crowder, H.: Validation of subgradient optimization. Math. Program. 6(1), 62–88 (1974) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jabr, R.A.: Robust self-scheduling under price uncertainty using conditional value-at-risk. IEEE Trans. Power Syst. 20(4), 1852–1858 (2005) CrossRefGoogle Scholar
  12. 12.
    Jabr, J., Baíllo, Á., Ventosa, M., García-Alcalde, A., Perán, F., Relaño, G.: A medium-term integrated risk management model for a hydrothermal generation company. IEEE Trans. Power Syst. 20(3), 1379–1388 (2005) CrossRefGoogle Scholar
  13. 13.
    Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, New York (1985) zbMATHGoogle Scholar
  14. 14.
    Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46(1), 105–122 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Krokhmal, P., Uryasev, S.: A sample-path approach to optimal position liquidation. Ann. Oper. Res. 152(1), 193–225 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Krokhmal, P., Uryasev, S., Zrazhevsky, G.: Risk management for hedge fund portfolios: A comparative analysis of linear portfolio rebalancing strategies. J. Altern. Invest. 5(1), 10–29 (2002) CrossRefGoogle Scholar
  17. 17.
    Krokhmal, P., Palmquist, J., Uryasev, S.: Portfolio optimization with conditional value-at-risk objective and constraints. J. Risk 4(2), 11–27 (2002) Google Scholar
  18. 18.
    Lim, C., Sherali, H.D.: A trust region target value method for optimizing nondifferentiable Lagrangian duals of linear programs. Math. Methods Oper. Res. 64(1), 33–53 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lim, C., Sherali, H.D.: Convergence and computational analyses for some variable target value and subgradient deflection methods. Comput. Optim. Appl. 34(3), 409–428 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Meucci, A.: Beyond Black-Litterman: Views on non-normal markets. Soc. Sci. Res. Netw. (2005)
  21. 21.
    Meucci, A.: Beyond Black-Litterman in practice: a five-step recipe to input views on non-normal markets. Soc. Sci. Res. Netw. (2005)
  22. 22.
    Polyak, B.T.: A general method of solving extremum problems. Sov. Math. 8(3), 593–597 (1967) zbMATHGoogle Scholar
  23. 23.
    Polyak, B.T.: Minimization of unsmooth functionals. U.S.S.R. Comput. Math. Math. Phys. 9(3), 14–39 (1969) CrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000) Google Scholar
  25. 25.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002) CrossRefGoogle Scholar
  26. 26.
    Scheuer, E.M., Stoller, D.S.: On the generation of normal random vectors. Technometrics 4(2), 278–281 (1962) zbMATHCrossRefGoogle Scholar
  27. 27.
    Shanno, D.F.: Conjugate gradient methods with inexact searches. Math. Oper. Res. 3(3), 244–256 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Sherali, H.D., Lim, C.: On embedding the volume algorithm in a variable target value method. Oper. Res. Lett. 32(5), 455–462 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Sherali, H.D., Lim, C.: Enhancing Lagrangian dual optimization for linear programs by obviating nondifferentiability. INFORMS J. Comput. 19(1), 3–13 (2007) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Sherali, H.D., Shetty, C.M.: On the generation of deep disjunctive cutting planes. Nav. Res. Logist. 27(3), 453–475 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sherali, H.D., Ulular, O.: Conjugate gradient methods using quasi-Newton updates with inexact line searches. J. Math. Anal. Appl. 150(2), 359–377 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Sherali, H.D., Choi, G., Tuncbilek, C.H.: A variable target value method for nondifferentiable optimization. Oper. Res. Lett. 26(1), 1–8 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Sherali, H.D., Choi, G., Ansari, Z.: Limited memory space dilation and reduction algorithms. Comput. Optim. Appl. 19(1), 55–77 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Uryasef, S.: New variable-metric algorithms for nondifferentiable optimization problems. J. Optim. Theory Appl. 71(2), 359–388 (1991) CrossRefMathSciNetGoogle Scholar
  35. 35.
    Uryasef, S.: Conditional value-at-risk: optimization algorithms and applications. Financ. Eng. News 14, 1–5 (2000) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Systems Engineering & Engineering ManagementUniversity of North Carolina at CharlotteCharlotteUSA
  2. 2.Grado Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  3. 3.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations