Mathematical Programming

, Volume 137, Issue 1–2, pp 167–198 | Cite as

Distributionally robust joint chance constraints with second-order moment information

  • Steve Zymler
  • Daniel Kuhn
  • Berç Rustem
Full Length Paper Series A


We develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove that this approximation is exact for robust individual chance constraints with concave or (not necessarily concave) quadratic constraint functions, and we demonstrate that the Worst-Case CVaR can be computed efficiently for these classes of constraint functions. Next, we study the Worst-Case CVaR approximation for joint chance constraints. This approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations. The tightness depends on a set of scaling parameters, which can be tuned via a sequential convex optimization algorithm. We show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the Worst-Case CVaR can be evaluated efficiently if the scaling parameters are kept constant. We evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations.

Mathematics Subject Classification (2010)

90C15 90C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alizadeh F., Goldfarb D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andrieu L., Henrion R., Römisch W.: A model for dynamic chance constraints in hydro power reservoir management. Eur. J. Oper. Res. 207(2), 579–589 (2010)zbMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Tal A., El Ghaoui L., Nemirovski A.: Robust Optimization. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  4. 4.
    Bertsimas D., Doan X., Natarajan K., Teo C.P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Calafiore G., Campi M.C.: The scenario approach to robust control design. IEEE Trans. Autom. Control 51(5), 742–753 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Calafiore G., El Ghaoui L.: Distributionally robust chance-constrained linear programs with applications. J. Optim. Theory Appl. 130(1), 1–22 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Calafiore G., Topcu U., El Ghaoui L.: Parameter estimation with expected and residual-at-risk criteria. Syst. Control Lett. 58(1), 39–46 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Charnes A., Cooper W.W., Symonds G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4(3), 235–263 (1958)CrossRefGoogle Scholar
  9. 9.
    Chen W., Sim M., Sun J., Teo C.P.: From CVaR to uncertainty set: implications in joint chance-constrained optimization. Oper. Res. 58(2), 470–485 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chen X., Sim M., Sun P.: A robust optimization perspective on stochastic programming. Oper. Res 55(6), 1058–1071 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. (2010)Google Scholar
  12. 12.
    Erdoǧan E., Iyengar G.: Ambiguous chance constrained problems and robust optimization. Math. Program. Series B 107, 37–61 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goh J., Sim M.: Robust optimization made easy with ROME. Oper. Res 59(4), 973–985 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Isii K.: The extrema of probability determined by generalized moments (i) bounded random variables. Ann. Inst. Stat. Math. 12(2), 119–134 (1960)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kuhn D., Wiesemann W., Georghiou A.: Primal and dual linear decision rules in stochastic and robust optimization. Math. Program 130(1), 177–209 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Löfberg, J.: YALMIP : a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei (2004)Google Scholar
  17. 17.
    Luedtke J., Ahmed S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Miller L.B., Wagner H.: Chance-constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)zbMATHCrossRefGoogle Scholar
  19. 19.
    Natarajan K., Pachamanova D., Sim M.: Constructing risk measures from uncertainty sets. Oper. Res. 57(5), 1129–1141 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Nemirovski A., Shapiro A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pagnoncelli B.K., Ahmed S., Shapiro A.: Sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl. 142(2), 399–416 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Pólik I., Terlaky T.: A survey of the S-lemma. SIAM Rev. 49(3), 371–481 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Rockafellar R.T., Uryasev S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2002)Google Scholar
  25. 25.
    Shapiro A.: On duality theory of conic linear problems. In: Goberna, M.A., Lopez, M.A. (eds) Semi-Infinite Programming: Recent Advances, Kluwer, Dordrecht (2001)Google Scholar
  26. 26.
    Shapiro A., Kleywegt A.J.: Minimax analysis of stochastic problems. Optim. Methods Softw 17(3), 523–542 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Toker, O., Ozbay, H.: On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: Proceedings of the American Control Conference, pp. 2525–2526. Seatle (1995)Google Scholar
  28. 28.
    Vandenberghe L., Boyd S., Comanor K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49, 52–64 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Zymler, S., Kuhn, D., Rustem, B.: Worst-case value-at-risk of non-linear portfolios. (2009)

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  1. 1.Department of ComputingImperial College LondonLondonUK

Personalised recommendations