Mathematical Programming

, Volume 151, Issue 1, pp 35–62 | Cite as

A distributionally robust perspective on uncertainty quantification and chance constrained programming

  • Grani A. Hanasusanto
  • Vladimir Roitch
  • Daniel Kuhn
  • Wolfram Wiesemann
Full Length Paper Series B


The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones.

Mathematics Subject Classification




This research was supported by the Swiss National Science Foundation under Grant BSCGI0_157733 and by EPSRC under Grant EP/I014640/1.


  1. 1.
    Ben-Tal, A., Ghaoui, L.El, Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ben-Tal, A., Den Hertog, D., De Waegenaere, A., Melenberg, B., Rennen, G.: Robust solutions of optimization problems affected by uncertain probabilities. Manag. Sci. 59(2), 341–357 (2013)CrossRefGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. A 88(3), 411–424 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, Philadelphia (2001)CrossRefGoogle Scholar
  5. 5.
    Bertsimas, D., Gupta, V., Kallus,N.: Data-driven robust optimization. Available on (2013)Google Scholar
  6. 6.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2004)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Calafiore, G.C., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1), 1–22 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Casella, G., Berger, R.L.: Statistical Inference, 2nd edn. Duxbury Thomson Learning, Pacific Grove, CA (2002)Google Scholar
  11. 11.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chen, X., Sim, M., Sun, P.: A robust optimization perspective on stochastic programming. Oper. Res. 55(6), 1058–1071 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 596–612 (2010)CrossRefMathSciNetGoogle Scholar
  14. 14.
    DeMiguel, V., Nogales, F.J.: Portfolio selection with robust estimation. Oper. Res. 57(3), 560–577 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Dharmadhikari, S.W., Joag-Dev, K.: Unimodality, Convexity, and Applications, Volume 27 of Probability and Mathematical Statistics. Academic Press, Waltham (1988)Google Scholar
  16. 16.
    Doan, X.V., Li, X., Natarajan, K.: Robustness to dependency in portfolio optimization using overlapping marginals. Available on optimization online (2013)Google Scholar
  17. 17.
    Doan, X.V., Natarajan, K.: On the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems. Oper. Res. 60(1), 138–149 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    El Ghaoui, L., Oks, M., Oustry, F.: Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51(4), 543–556 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. B 107(1–2), 37–61 (2006)CrossRefzbMATHGoogle Scholar
  20. 20.
    Gauss, C.F.: Theoria combinationis observationum erroribus minimis obnoxiae, pars prior. Comment. Soc. Reg. Sci. Gott. Recent. 33, 321–327 (1821)Google Scholar
  21. 21.
    Han, S., Tao, M., Topcu, U., Owhadi, H., Murray, R. M.: Convex optimal uncertainty quantification. Available on, (2013)Google Scholar
  22. 22.
    Hanasusanto, G. A., Roitch, V., Kuhn, D., Wiesemann,W.: Ambiguous joint chance constraints with conic dispersion measures. Working Paper, Imperial College London and École Polytechnique Fédérale de Lausanne 2015Google Scholar
  23. 23.
    Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: A distributionally robust perspective on uncertainty quantification and chance constrained programming. Technical Report, Imperial College London and École Polytechnique Fédérale de Lausanne (2015)Google Scholar
  24. 24.
    Hu Z., Hong,L. J.: Kullback-Leibler divergence constrained distributionally robust optimization. Available on optimization online (2012)Google Scholar
  25. 25.
    Hu, Z., Hong, L. J., So,A. M.-C.: Ambiguous probabilistic programs. Available on optimization online (2013)Google Scholar
  26. 26.
    Huber, P.J.: Robust estimation of a location parameter. Ann. Stat. 53(1), 73–101 (1964)CrossRefGoogle Scholar
  27. 27.
    Jasour, A., Aybat, N. S., Lagoa, C.: Semidefinite programming for chance optimization over semialgebraic sets. Available on (2014)Google Scholar
  28. 28.
    Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Available on optimization online (2012)Google Scholar
  29. 29.
    Korski, J., Pfeuffer, F., Klamroth, K.: Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Methods Oper. Res. 66(3), 373–407 (2007)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Lam, S.-W., Ng, T.S., Sim, M., Song, J.-H.: Multiple objectives satisficing under uncertainty. Oper. Res. 61(1), 214–227 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Mohajerin Esfahani, P., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Working Paper, École Polytechnique Fédérale de Lausanne (2015)Google Scholar
  32. 32.
    Natarajan, K., Pachamanova, D., Sim, M.: Incorporating asymmetric distributional information in robust value-at-risk optimization. Manag. Sci. 54(3), 573–585 (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Owhadi, H., Scovel, C., Sullivan, T.J., McKerns, M., Ortiz, M.: Optimal uncertainty quantification. SIAM Rev. 55(2), 271–345 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Pflug, G., Pichler, A., Wozabal, D.: The \(1/n\) investment strategy is optimal under high model ambiguity. J. Bank. Finance 36(2), 410–417 (2012)CrossRefGoogle Scholar
  36. 36.
    Popescu, I.: An SDP approach to optimal moment bounds for convex classes of distributions. Math. Oper. Res. 50(3), 632–657 (2005)CrossRefGoogle Scholar
  37. 37.
    Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, New York (1991)zbMATHGoogle Scholar
  38. 38.
    Shapiro, A.: On duality theory of conic linear problems. In: Semi-infinite Programming, chapter 7, pp 135–165. Kluwer Academic Publishers (2001)Google Scholar
  39. 39.
    Shapiro, A., Kleywegt, A.: Minimax analysis of stochastic problems. Optim. Methods Softw. 17(3), 523–542 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Sun, H., Xu, H.: Asymptotic convergence analysis for distributional robust optimization and equilibrium problems. Available on optimization online (2013)Google Scholar
  41. 41.
    Van Parys, B.P.G., Goulart, P.J., Kuhn, D.: Generalized Gauss inequalities via semidefinite programming. Math. Program. A (2015) (in press)Google Scholar
  42. 42.
    Van Parys, B.P.G., Kuhn, D., Goulart, P.J., Morari, M.: Distributionally robust control of constrained stochastic systems. Available on optimization online (2013)Google Scholar
  43. 43.
    Vandenberghe, L., Boyd, S., Comanor, K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49(1), 52–64 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Wiesemann, W., Kuhn, D., Rustem, B.: Robust Markov decision processes. Math. Oper. Res. 38(1), 153–183 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Xu, H., Caramanis, C., Mannor, S.: Optimization under probabilistic envelope constraints. Oper. Res. 60(3), 682–699 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Yanıkoğlu, İ., Den Hertog, D.: Safe approximations of ambiguous chance constraints using historical data. INFORMS J. Comput. 25(4), 666–681 (2013)CrossRefMathSciNetGoogle Scholar
  48. 48.
    Žáčková, J.: On minimax solutions of stochastic linear programming problems. Čas. Pěst. Mat. 91(4), 423–430 (1966)zbMATHGoogle Scholar
  49. 49.
    Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. A 137(1–2), 167–198 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Zymler, S., Kuhn, D., Rustem, B.: Worst-case value-at-risk of non-linear portfolios. Manag. Sci. 59(1), 172–188 (2013)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Grani A. Hanasusanto
    • 1
  • Vladimir Roitch
    • 1
  • Daniel Kuhn
    • 2
  • Wolfram Wiesemann
    • 3
  1. 1.Department of ComputingImperial College LondonLondonUnited Kingdom
  2. 2.Risk Analytics and Optimization ChairÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.Imperial College Business SchoolImperial College LondonLondonUnited Kingdom

Personalised recommendations