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Optimized Bonferroni approximations of distributionally robust joint chance constraints

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Abstract

A distributionally robust joint chance constraint involves a set of uncertain linear inequalities which can be violated up to a given probability threshold \(\epsilon \), over a given family of probability distributions of the uncertain parameters. A conservative approximation of a joint chance constraint, often referred to as a Bonferroni approximation, uses the union bound to approximate the joint chance constraint by a system of single chance constraints, one for each original uncertain constraint, for a fixed choice of violation probabilities of the single chance constraints such that their sum does not exceed \(\epsilon \). It has been shown that, under various settings, a distributionally robust single chance constraint admits a deterministic convex reformulation. Thus the Bonferroni approximation approach can be used to build convex approximations of distributionally robust joint chance constraints. In this paper we consider an optimized version of Bonferroni approximation where the violation probabilities of the individual single chance constraints are design variables rather than fixed a priori. We show that such an optimized Bonferroni approximation of a distributionally robust joint chance constraint is exact when the uncertainties are separable across the individual inequalities, i.e., each uncertain constraint involves a different set of uncertain parameters and corresponding distribution families. Unfortunately, the optimized Bonferroni approximation leads to NP-hard optimization problems even in settings where the usual Bonferroni approximation is tractable. When the distribution family is specified by moments or by marginal distributions, we derive various sufficient conditions under which the optimized Bonferroni approximation is convex and tractable. We also show that for moment based distribution families and binary decision variables, the optimized Bonferroni approximation can be reformulated as a mixed integer second-order conic set. Finally, we demonstrate how our results can be used to derive a convex reformulation of a distributionally robust joint chance constraint with a specific non-separable distribution family.

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References

  1. Ahmed, S.: Convex relaxations of chance constrained optimization problems. Optim. Lett. 8(1), 1–12 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Ahmed, S., Luedtke, J., Song, Y., Xie, W.: Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs. Math. Program. 162(1–2), 51–81 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Bonferroni, C.E.: Teoria statistica delle classi e calcolo delle probabilità. Libreria internazionale Seeber, Florence (1936)

    MATH  Google Scholar 

  4. Bukszár, J., Mádi-Nagy, G., Szántai, T.: Computing bounds for the probability of the union of events by different methods. Ann. Oper. Res. 201(1), 63–81 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Calafiore, G.C., Campi, M.C.: The scenario approach to robust control design. IEEE Trans. Autom. Control 51(5), 742–753 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Calafiore, G.C., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1), 1–22 (2006)

    MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. Cheng, J., Delage, E., Lisser, A.: Distributionally robust stochastic knapsack problem. SIAM J. Optim. 24(3), 1485–1506 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Cheng, J., Gicquel, C., Lisser, A.: Partial sample average approximation method for chance constrained problems. Optim. Lett. 13(4), 657–672 (2018). https://doi.org/10.1007/s11590-018-1300-8

    Article  MathSciNet  MATH  Google Scholar 

  10. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)

    MathSciNet  MATH  Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. Esfahani, P.M., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Math. Program. 171(1–2), 115–166 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Fréchet, M.: Généralisation du théoreme des probabilités totales. Fundamenta mathematicae 1(25), 379–387 (1935)

    MATH  Google Scholar 

  14. Gao, R., Kleywegt, A.J.: Distributionally robust stochastic optimization with Wasserstein distance. arXiv preprint arXiv:1604.02199. (2016)

  15. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math. Program. 151, 35–62 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: Ambiguous joint chance constraints under mean and dispersion information. Oper. Res. 65(3), 751–767 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Hunter, D.: An upper bound for the probability of a union. J. Appl. Probab. 13(3), 597–603 (1976)

    MathSciNet  MATH  Google Scholar 

  19. Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Math. Program. 158, 291–327 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Kataoka, S.: A stochastic programming model. Econometrica: J. Econom. Soc. 31, 181–196 (1963)

    MathSciNet  MATH  Google Scholar 

  21. Kuai, H., Alajaji, F., Takahara, G.: A lower bound on the probability of a finite union of events. Discrete Math. 215(1–3), 147–158 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Lagoa, C.M., Li, X., Sznaier, M.: Probabilistically constrained linear programs and risk-adjusted controller design. SIAM J. Optim. 15(3), 938–951 (2005)

    MathSciNet  MATH  Google Scholar 

  23. Li, B., Jiang, R., Mathieu, J.L.: Ambiguous risk constraints with moment and unimodality information. Math. Program. 173(1), 151–192 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122, 247–272 (2010)

    MathSciNet  MATH  Google Scholar 

  26. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10(1), 147–175 (1976)

    MATH  Google Scholar 

  27. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)

    MathSciNet  MATH  Google Scholar 

  28. Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty, pp. 3–47. Springer, London (2006)

    Google Scholar 

  29. Prékopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming, pp. 113–138 (1970)

  30. Prékopa, A.: Stochastic Programming. Springer, Berlin (1995)

    MATH  Google Scholar 

  31. Prékopa, A.: On the concavity of multivariate probability distribution functions. Oper. Res. Lett. 29(1), 1–4 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Prékopa, A.: Probabilistic programming. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10, pp. 267–351. Elsevier (2003)

  33. Prékopa, A., Gao, L.: Bounding the probability of the union of events by aggregation and disaggregation in linear programs. Discrete Appl. Math. 145(3), 444–454 (2005)

    MathSciNet  MATH  Google Scholar 

  34. Rockafellar, R., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Google Scholar 

  35. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317. Springer, Berlin (1998)

    MATH  Google Scholar 

  36. Rüschendorf, L.: Sharpness of Fréchet-bounds. Probab. Theory Relat. Fields 57(2), 293–302 (1981)

    MATH  Google Scholar 

  37. Shapiro, A., Kleywegt, A.: Minimax analysis of stochastic problems. Optim. Methods Softw. 17(3), 523–542 (2002)

    MathSciNet  MATH  Google Scholar 

  38. Song, Y., Luedtke, J.R., Küçükyavuz, S.: Chance-constrained binary packing problems. INFORMS J. Comput. 26(4), 735–747 (2014)

    MathSciNet  MATH  Google Scholar 

  39. Wagner, M.R.: Stochastic 0–1 linear programming under limited distributional information. Oper. Res. Lett. 36(2), 150–156 (2008)

    MathSciNet  MATH  Google Scholar 

  40. Wets, R.: Stochastic programming: solution techniques and approximation schemes. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming: State-of-the-Art 1982, pp. 566–603. Springer, Berlin (1983)

    Google Scholar 

  41. Wets, R.: Stochastic programming. In: Nemhauser, G.L., Rinooy Kan, A.H.G., Todd, M.J. (eds.) Handbooks in Operations Research and Management Science, vol. 1, pp. 573–629. Elsevier (1989)

  42. Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Xie, W., Ahmed, S.: Distributionally robust chance constrained optimal power flow with renewables: a conic reformulation. IEEE Trans. Power Syst. 33(2), 1860–1867 (2018)

    Google Scholar 

  44. Xie, W., Ahmed, S.: On deterministic reformulations of distributionally robust joint chance constrained optimization problems. SIAM J. Optim. 28(2), 1151–1182 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Zhang, Y., Jiang, R., Shen, S.: Ambiguous chance-constrained binary programs under mean-covariance information. SIAM J. Optim. 28(4), 2922–2944 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137, 167–198 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research has been supported in part by the National Science Foundation Awards #1633196 and #1662774.

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Authors and Affiliations

  1. Virginia Tech, Blacksburg, VA, USA

    Weijun Xie

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Shabbir Ahmed

  3. University of Michigan, Ann Arbor, MI, USA

    Ruiwei Jiang

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  1. Weijun Xie
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  2. Shabbir Ahmed
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  3. Ruiwei Jiang
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Correspondence to Ruiwei Jiang.

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Xie, W., Ahmed, S. & Jiang, R. Optimized Bonferroni approximations of distributionally robust joint chance constraints. Math. Program. 191, 79–112 (2022). https://doi.org/10.1007/s10107-019-01442-8

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