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Data-driven distributionally robust chance-constrained optimization with Wasserstein metric

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Abstract

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem.

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Notes

  1. In an abuse of notation, we use the same symbol \(\xi \) in chance-constrained models involving either random right-hand side or random technology vector models, although \(\xi \) represents a scalar in the first case and a vector of dimension M in the second one.

  2. We shall interchangeably use the terms atoms and realizations of uncertain variables \(\xi \).

  3. In an abuse of notation, we use the same symbols \(\nu \), q and \(\alpha \) here as in Theorem 2.

References

  1. Blanchet, J., Chen, L., Zhou, X.Y.: Distributionally robust mean–variance portfolio selection with Wasserstein distance. Working paper: available at Optimization Online (2018)

  2. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  3. Calafiore, G.C.: Ambiguous risk measures and optimal robust portfolios. SIAM J. Optim. 18(3), 853–877 (2007)

    Article  MathSciNet  Google Scholar 

  4. Chen, Z., Kuhn, D., Wiesemann, W.: Data-driven chance constrained programs over Wasserstein balls. Working paper: available at Optimization Online (2018)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Duan, C., Fang, W., Jiang, L., Yao, L., Liu, J.: Distributionally robust chance-constrained voltage-concerned dc-opf with Wasserstein metric. Working paper. arXiv: 1706.05538v1 (2017)

  8. Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107(1–2), 37–61 (2006)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

  10. Gao, R., Kleywegt, A.J.: Distributionally robust stochastic optimization with Wasserstein distance. Working paper: available at Optimization Online (2016)

  11. 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)

    Article  MathSciNet  Google Scholar 

  12. Hu, Z., Hong, L.J.: Kullback-Leibler divergence constrained distributionally robust optimization. Working paper: available at Optimization Online (2013)

  13. IBM: IBM big data hub—the four v’s of big data. http://www.ibmbigdatahub.com/sites/default/files/infographic_file/4-Vs-of-big-data.jpg (2017)

  14. Ji, R., Lejeune, M.: Data-driven optimization of reward-risk ratio measures. INFORMS J. Comput. (2020) (in press)

  15. Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Math. Program. 158, 291–327 (2015)

    Article  MathSciNet  Google Scholar 

  16. Jiang, R., Guan, Y., Watson, J.P.: Risk-averse stochastic unit commitment with incomplete information. IIE Trans. 48(9), 838–854 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Pflug, G.C., Pohl, M.: A review on ambiguity in stochastic portfolio optimization. Set-Valued Var. Anal. 26, 1–25 (2017)

    MathSciNet  Google Scholar 

  19. Pflug, G.C., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7(4), 435–442 (2007)

    Article  MathSciNet  Google Scholar 

  20. Pflug, G.C., Pichler, A., Wozabal, D.: The 1/N investment strategy is optimal under high model ambiguity. J. Bank. Finance 36(2), 410–417 (2012)

    Article  Google Scholar 

  21. Postek, K., Den Hertog, D., Melenberg, B.: Computationally tractable counterparts of distributionally robust constraints on risk measures. SIAM Rev. 58(4), 603–650 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Xie, W.: On distributionally robust chance constrained programs with Wasserstein distance. Math. Program. 1–41 (2018). https://doi.org/10.1007/s10107-019-01445-5

  24. Xie, W., Ahmed, S.: Bicriteria approximation of chance-constrained covering problems. Oper. Res. 68(2), 516–533 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Yang, W., Xu, H.: Distributionally robust chance constraints for non-linear uncertainties. Math. Program. 155(1–2), 231–265 (2016)

    Article  MathSciNet  Google Scholar 

  26. Zhao, C., Guan, Y.: Data-driven risk-averse stochastic optimization with Wasserstein metric. Oper. Res. Lett. 46(2), 262–267 (2018)

    Article  MathSciNet  Google Scholar 

  27. Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137(1–2), 167–198 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

M. Lejeune acknowledges the partial support provided by the Office of Naval Research [Grant N000141712420].

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Authors and Affiliations

  1. Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA, USA

    Ran Ji

  2. Department of Decision Sciences, George Washington University, Washington, DC, USA

    Miguel A. Lejeune

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  1. Ran Ji
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  2. Miguel A. Lejeune
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Correspondence to Ran Ji.

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Ji, R., Lejeune, M.A. Data-driven distributionally robust chance-constrained optimization with Wasserstein metric. J Glob Optim 79, 779–811 (2021). https://doi.org/10.1007/s10898-020-00966-0

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  • DOI: https://doi.org/10.1007/s10898-020-00966-0

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