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Scenario reduction for stochastic programs with Conditional Value-at-Risk

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Abstract

In this paper we discuss scenario reduction methods for risk-averse stochastic optimization problems. Scenario reduction techniques have received some attention in the literature and are used by practitioners, as such methods allow for an approximation of the random variables in the problem with a moderate number of scenarios, which in turn make the optimization problem easier to solve. The majority of works for scenario reduction are designed for classical risk-neutral stochastic optimization problems; however, it is intuitive that in the risk-averse case one is more concerned with scenarios that correspond to high cost. By building upon the notion of effective scenarios recently introduced in the literature, we formalize that intuitive idea and propose a scenario reduction technique for stochastic optimization problems where the objective function is a Conditional Value-at-Risk. Numerical results presented with problems from the literature illustrate the performance of the method and indicate the cases where we expect it to perform well.

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Notes

  1. Models with CVaR have also been used in the context of multistage stochastic programs, but we do not review them here as we focus on the two-stage case in this paper.

  2. By “reducing” the number of scenarios we mean finding another distribution \(\hat{P}\) such that the support of \(\hat{P}\) is strictly contained in \(\Xi \) (the support of \(\hat{P}\) consists of those scenarios \(\xi _i\) for which \(\hat{P}_i>0\)).

  3. In the most general case W and q are random as well, but for our purposes we assume that these quantities are deterministic.

  4. An implementation of ZEBRA can be found in https://sites.google.com/site/sergiogarciaquiles/the-p-median-facility-location-problem.

References

  1. Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Program. 106(3), 433–446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Altenstedt, F.: Aspects on Asset Liability Management Via Stochastic Programming. Chalmers University of Technology, Gothenburg (2003)

    Google Scholar 

  3. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–227 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (2011)

    Book  MATH  Google Scholar 

  5. Cotton, T.G., Ntaimo, L.: Computational study of decomposition algorithms for mean-risk stochastic linear programs. Math. Program. Comput. 7(4), 471–499 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czyzyk, J., Mesnier, M.P., Moré, J.J.: The neos server. IEEE J. Comput. Sci. Eng. 5(3), 68–75 (1998)

    Article  Google Scholar 

  7. Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

    Book  MATH  Google Scholar 

  8. Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci. 1(3–4), 197–206 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolan, E.D.: The neos server 4.0 administrative guide (Technical Memorandum NO. ANL/MCS-TM-250). Mathematics and Computer Science Division, Argonne National Laboratory (2001)

  10. Dupačová, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming. Math. Program. 95(3), 493–511 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dupačová, J., Consigli, G., Wallace, S.W.: Scenarios for multistage stochastic programs. Ann. Oper. Res. 100, 25–53 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dupačová, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming: An approach using probability metrics. Math. Program. 95, 493–511 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eichhorn, A., Römisch, W.: Stability of multistage stochastic programs incorporating polyhedral risk measures. Optimization 57(2), 295–318 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Espinoza, D., Moreno, E.: A primal-dual aggregation algorithm for minimizing conditional value-at-risk in linear programs. Comput. Optim. Appl. 59(3), 617–638 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fairbrother, J., Turner, A., Wallace, S.: Scenario generation for stochastic programs with tail risk measures (2015). arXiv preprint arXiv:1511.03074

  16. García, S., Labbé, M., Marín, A.: Solving large p-median problems with a radius formulation. INFORMS J. Comput. 23(4), 546–556 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. García-Bertrand, R., Mínguez, R.: Iterative scenario based reduction technique for stochastic optimization using conditional value-at-risk. Optim. Eng. 15(2), 355–380 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gropp, W., Moré, J.J.: Optimization environments and the neos server. In: Buhman, Martin D., Iserles, Arieh (eds.) Approximation Theory and Optimization, pp. 167–182. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  19. Guigues, V., Krätschmer, V., Shapiro, A.: Statistical inference and hypotheses testing of risk averse stochastic programs (2016). arXiv preprint arXiv:1603.07384

  20. Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24, 187–206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Heitsch, H., Römisch, W.: Scenario tree modeling for multistage stochastic programs. Math. Program. 118, 371–406 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Homem-de-Mello, T., Bayraksan, G.: Monte Carlo sampling-based methods for stochastic optimization. Surv. Oper. Res. Manag. Sci. 19, 56–85 (2014)

    MathSciNet  Google Scholar 

  23. Hoyland, K., Kaut, M., Wallace, S.W.: A heuristic for moment-matching scenario generation. Comput. Optim. Appl. 24, 169–185 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoyland, K., Wallace, S.W.: Generating scenario trees for multistage decision problems. Manag. Sci. 47(2), 295–307 (2001)

    Article  MATH  Google Scholar 

  25. Linderoth, J., Shapiro, A., Wright, S.: The empirical behavior of sampling methods for stochastic programming. Ann. Oper. Res. 142(1), 215–241 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Louveaux, F., Smeers, Y.: Optimal investments for electricity generation: a stochastic model and a test problem. In: Ermoliev, Y., Wets, R.J.-B. (eds.) Numericaltechniques for Stochastic Optimization Problems, pp. 445–452. Springer, Berlin (1988)

    Google Scholar 

  27. McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979)

    MathSciNet  MATH  Google Scholar 

  28. Mehrotra, S., Papp, D.: A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization. SIAM J. Optim. 24(4), 1670–1697 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miller, N., Ruszczynski, A.: Risk-averse two-stage stochastic linear programming: modeling and decomposition. Oper. Res. 59, 125–132 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Noyan, N.: Risk-averse two-stage stochastic programming with an application to disaster management. Comput. Oper. Res. 39(3), 541–559 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pflug, G.C.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. Ser. B 89(2), 251–271 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pflug, G.C., Pichler, A.: Approximations for probability distributions and stochastic optimization problems. In: Bertocchi, M., Consigli, G., Dempster, M.A.H. (eds.) Stochastic Optimization Methods in Finance and Energy, pp. 343–387. Springer, New York (2011)

    Chapter  Google Scholar 

  33. Pineda, S., Conejo, A.: Scenario reduction for risk-averse electricity trading. IET Gener. Transm. Distrib. 4(6), 694–705 (2010)

    Article  Google Scholar 

  34. Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models, vol. 269. Wiley, Hoboken (1991)

    MATH  Google Scholar 

  35. Rahimian, H., Bayraksan, G., Homem-de Mello, T.: Identifying effective scenarios in distributionally robust stochastic programs with total variation distance. Math. Program. (2018). https://doi.org/10.1007/s10107-017-1224-6

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

    Article  Google Scholar 

  37. Rockafellar, R.T., Wets, R.J.: Variational Analysis: Grundlehren der mathematischen wissenschaften. Springer, Berlin (1998)

    Book  Google Scholar 

  38. Römisch, W., Wets, R.-B.: Stability of \(\varepsilon \)-approximate solutions to convex stochastic programs. SIAM J. Optim. 18(3), 961–979 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shapiro, A.: Inference of statistical bounds for multistage stochastic programming problems. Math. Meth. Oper. Res. 58, 57–68 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, 2nd edn. SIAM, Philadelphia (2014)

    MATH  Google Scholar 

  41. Wallace, S.W., Ziemba, W.T.: Applications of Stochastic Programming, vol. 5. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

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Acknowledgements

We thank the anonymous referees for their constructive comments which helped improve the presentation of our results. This work has been supported by FONDECYT 1171145, Chile.

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  1. School of Business, Universidad Adolfo Ibañez, Santiago, Chile

    Sebastián Arpón, Tito Homem-de-Mello & Bernardo Pagnoncelli

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  1. Sebastián Arpón
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Correspondence to Tito Homem-de-Mello.

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Arpón, S., Homem-de-Mello, T. & Pagnoncelli, B. Scenario reduction for stochastic programs with Conditional Value-at-Risk. Math. Program. 170, 327–356 (2018). https://doi.org/10.1007/s10107-018-1298-9

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