α-Conservative approximation for probabilistically constrained convex programs

  • Yuichi Takano
  • Jun-ya GotohEmail author


In this paper, we address an approximate solution of a probabilistically constrained convex program (PCCP), where a convex objective function is minimized over solutions satisfying, with a given probability, convex constraints that are parameterized by random variables. In order to approach to a solution, we set forth a conservative approximation problem by introducing a parameter α which indicates an approximate accuracy, and formulate it as a D.C. optimization problem.

As an example of the PCCP, the Value-at-Risk (VaR) minimization is considered under the assumption that the support of the probability of the associated random loss is given by a finitely large number of scenarios. It is advantageous in solving the D.C. optimization that the numbers of variables and constraints are independent of the number of scenarios, and a simplicial branch-and-bound algorithm is posed to find a solution of the D.C. optimization. Numerical experiments demonstrate the following: (i) by adjusting a parameter α, the proposed problem can achieve a smaller VaR than the other convex approximation approaches; (ii) when the number of scenarios is large, a typical 0-1 mixed integer formulation for the VaR minimization cannot be solved in a reasonable time and the improvement of the incumbent values is slow, whereas the proposed method can achieve a good solution at an early stage of the algorithm.


Chance constraint D.C. optimization Branch-and-bound Value-at-risk minimization Probabilistically constrained program 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benati, S., Rizzi, R.: A mixed integer linear programming formulation of the optimal mean/value-at-risk portfolio problem. Eur. J. Oper. Res. 176, 423–434 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertsimas, D., Sim, M.: Price of robustness. Oper. Res. 52, 35–53 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Calafiore, G., Campi, M.C.: Decision making in an uncertain environment: the scenario-based optimization approach. In: Andrysek, J., Karny, M., Kracik, J. (eds.) Multiple Participant Decision Making, pp. 99–111. Advanced Knowledge International (2004) Google Scholar
  6. 6.
    Calafiore, G., Campi, M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4, 235–263 (1958) CrossRefGoogle Scholar
  8. 8.
    Dentcheva, D.: Optimization models with probabilistic constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty, pp. 49–97. Springer, London (2006) CrossRefGoogle Scholar
  9. 9.
    El Ghaoui, L., Oks, M., Oustry, F.: Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51, 543–556 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gaivoronski, A.A., Pflug, G.: Value-at-risk in portfolio optimization: properties and computational approach. J. Risk 7, 1–31 (2005) Google Scholar
  11. 11.
    Gilli, M., Këllezi, E., Hysi, H.: A data-driven optimization heuristic for downside risk minimization. J. Risk 8, 1–19 (2006) Google Scholar
  12. 12.
  13. 13.
    Konno, H., Thach, P.T., Tuy, H.: Optimization on Low Rank Nonconvex Structures. Kluwer Academic, Dordrecht (1997) zbMATHGoogle Scholar
  14. 14.
    Larsen, N., Mausser, H., Uryasev, S.: Algorithms for optimization of value-at-risk. In: Pardalos, P., Tsitsiringos, V.K. (eds.) Financial Engineering, e-Commerce and Supply Chain, pp. 129–157. Kluwer Academic, Dordrecht (2002) Google Scholar
  15. 15.
    Miller, L.B., Wagner, H.: Chance constrained programming with joint constraints. Oper. Res. 13, 930–945 (1965) zbMATHCrossRefGoogle Scholar
  16. 16.
    Natarajan, K., Pachamanova, D., Sim, M.: Incorporating asymmetric distributional information in robust value-at-risk optimization. Manag. Sci. 54, 573–585 (2008) zbMATHCrossRefGoogle Scholar
  17. 17.
    Nemirovski, A.: On tractable approximations of randomly perturbed convex constraints. In: Proceedings of the 42nd IEEE Conference on Decision and Control, vol. 3, pp. 2419–2422 (2003) Google Scholar
  18. 18.
    Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pang, J.S., Leyffer, S.: On the global minimization of the value-at-risk. Optim. Methods Softw. 19, 611–631 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Prékopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming, pp. 113–138. Princeton University Press, Princeton (1970) Google Scholar
  21. 21.
    Puelz, A.: Value-at-risk based portfolio optimization. In: Uryasev, S., Pardalos, P.M. (eds.) Stochastic Optimization: Algorithms and Applications, pp. 279–302. Kluwer Academic, Dordrecht (2001) Google Scholar
  22. 22.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443–1471 (2002) CrossRefGoogle Scholar
  23. 23.
    Tuy, H.: Convex Analysis and Global Optimization. Kluwer Academic, Dordrecht (1998) zbMATHGoogle Scholar
  24. 24.
    Verma, A.: VaR optimal portfolios. A global optimization approach. In: Workshop on Optimization in Finance, Coimbra (2005) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukuba, IbarakiJapan
  2. 2.Department of Industrial and Systems EngineeringChuo UniversityTokyoJapan

Personalised recommendations