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Mathematical Programming

, Volume 173, Issue 1–2, pp 151–192 | Cite as

Ambiguous risk constraints with moment and unimodality information

  • Bowen Li
  • Ruiwei JiangEmail author
  • Johanna L. Mathieu
Full Length Paper Series A

Abstract

Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations.

Keywords

Ambiguity Chance constraints Conditional Value-at-Risk Second-order cone representation Separation Golden section search 

Mathematics Subject Classification

90C15 Stochastic programming 90C22 Semidefinite programming 90C34 Semi-infinite programming 

Notes

Acknowledgements

This research has been supported in part by the National Science Foundation (NSF) under Grants CMMI-1555983 and CCF-1442495.

References

  1. 1.
    Ahmed, S., Papageorgiou, D.: Probabilistic set covering with correlations. Oper. Res. 61(2), 438–452 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bergen, A.R., Vittal, V.: Power Systems Analysis, 2nd edn. Prentice Hall, Upper Saddle River (1999)Google Scholar
  4. 4.
    Bertsimas, D., Doan, X.V., Natarajan, K., Teo, C.-P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bienstock, D., Chertkov, M., Harnett, S.: Chance-constrained optimal power flow: risk-aware network control under uncertainty. SIAM Rev. 56(3), 461–495 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bookbinder, J.H., Tan, J.-Y.: Strategies for the probabilistic lot-sizing problem with service-level constraints. Manag. Sci. 34(9), 1096–1108 (1988)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Calafiore, G., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1), 1–22 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Charnes, A., Cooper, W., Symonds, G.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4(3), 235–263 (1958)Google Scholar
  9. 9.
    Cheng, J., Delage, E., Lisser, A.: Distributionally robust stochastic knapsack problem. SIAM J. Optim. 24(3), 1485–1506 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dharmadhikari, S., Joag-Dev, K.: Unimodality, Convexity, and Applications. Elsevier, Amsterdam (1988)zbMATHGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107(1), 37–61 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Esfahani, P. M., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Available on optimization-online. http://www.optimization-online.org/DB_FILE/2015/05/4899.pdf (2015)
  15. 15.
    Gade, D., Küçükyavuz, S.: Formulations for dynamic lot sizing with service levels. Nav. Res. Logist. 60(2), 87–101 (2013)MathSciNetGoogle Scholar
  16. 16.
    Gómez-Expósito, A., Conejo, A.J., Cañizares, C.: Electric Energy Systems: Analysis and Operation. CRC Press, Boca Raton (2008)Google Scholar
  17. 17.
    Hanasusanto, G. A.: Decision making under uncertainty: robust and data-driven approaches. Ph.D. thesis, Imperial College London (2015)Google Scholar
  18. 18.
    Hanasusanto, G. A., Roitch, V., Kuhn, D., Wiesemann, W.: Ambiguous joint chance constraints under mean and dispersion information. Available on optimization-online. http://www.optimization-online.org/DB_FILE/2015/11/5199.pdf (2015)
  19. 19.
    Henrion, R., Li, P., Möller, A., Steinbach, M.C., Wendt, M., Wozny, G.: Stochastic optimization for operating chemical processes under uncertainty. In: Grötschel, M., Krunke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems, pp. 457–478. Springer, Berlin (2001)Google Scholar
  20. 20.
    Henrion, R., Möller, A.: Optimization of a continuous distillation process under random inflow rate. Comput. Math. Appl. 45(1), 247–262 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Math. Program. 158(1–2), 291–327 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Li, B., Jiang, R., Mathieu, J. L.: Distributionally robust risk-constrained optimal power flow using moment and unimodality information. In: 55th IEEE Conference on Decision and Control (CDC). IEEE (2016)Google Scholar
  23. 23.
    Miller, B., Wagner, H.: Chance constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)zbMATHGoogle Scholar
  24. 24.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, Hoboken (1999)zbMATHGoogle Scholar
  25. 25.
    Ozturk, U., Mazumdar, M., Norman, B.: A solution to the stochastic unit commitment problem using chance constrained programming. IEEE Trans. Power Syst. 19(3), 1589–1598 (2004)Google Scholar
  26. 26.
    Popescu, I.: A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Math. Oper. Res. 30(3), 632–657 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Popescu, I.: Robust mean–covariance solutions for stochastic optimization. Oper. Res. 55(1), 98–112 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)Google Scholar
  29. 29.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)Google Scholar
  30. 30.
    Scarf, H.: A min-max solution of an inventory problem. In: Arrow, K., Karlin, S., Scarf, H. (eds.) Studies in the Mathematical Theory of Inventory and Production, pp. 201–209. Stanford University Press, Palo Alto (1958)Google Scholar
  31. 31.
    Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.Á., López M.A. (eds.) Semi-Infinite Programming, pp. 135–165. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  32. 32.
    Shapiro, A., Kleywegt, A.: Minimax analysis of stochastic problems. Optim. Methods Softw. 17(3), 523–542 (2002)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Stellato, B.: Data-driven chance constrained optimization. Master’s thesis, ETH Zürich (2014)Google Scholar
  35. 35.
    Van Parys, B. P. G., Goulart, P. J., Kuhn, D.: Generalized Gauss inequalities via semidefinite programming. Math. Program. 156(1–2), 271–302 (2015a)Google Scholar
  36. 36.
    Van Parys, B. P. G., Goulart, P. J., Morari, M.: Distributionally robust expectation inequalities for structured distributions. Available on optimization-online. http://www.optimization-online.org/DB_FILE/2015/05/4896.pdf (2015b)
  37. 37.
    Vandenberghe, L., Boyd, S., Comanor, K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49(1), 52–64 (2007)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Vrakopoulou, M., Margellos, K., Lygeros, J., Andersson, G.: A probabilistic framework for reserve scheduling and \(N-1\) security assessment of systems with high wind power penetration. IEEE Trans. Power Syst. 28(4), 3885–3896 (2013)Google Scholar
  39. 39.
    Wagner, M.: Stochastic 0–1 linear programming under limited distributional information. Oper. Res. Lett. 36(2), 150–156 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Wang, Q., Guan, Y., Wang, J.: A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output. IEEE Trans. Power Syst. 27(1), 206–215 (2012)MathSciNetGoogle Scholar
  41. 41.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Zimmerman, R.D., Murillo-Sánchez, C.E., Thomas, R.J.: MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2011)Google Scholar
  43. 43.
    Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137(1–2), 167–198 (2013a)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Zymler, S., Kuhn, D., Rustem, B.: Worst-case value at risk of nonlinear portfolios. Manag. Sci. 59(1), 172–188 (2013b)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of MichiganAnn ArborUSA
  2. 2.Department of Industrial and Operations EngineeringUniversity of MichiganAnn ArborUSA

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