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Mathematical Programming

, Volume 174, Issue 1–2, pp 99–127 | Cite as

Generalized conditioning based approaches to computing confidence intervals for solutions to stochastic variational inequalities

  • Michael LammEmail author
  • Shu Lu
Full Length Paper Series B
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Abstract

Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals.

Mathematics Subject Classification

90C33 90C15 65K10 62F25 

Notes

Acknowledgements

Research of Michael Lamm and Shu Lu is supported by National Science Foundation under the Grants DMS-1109099 and DMS-1407241. Research of Michael Lamm took place during his graduate study in the Department of Statistics and Operations at the University of North Carolina at Chapel Hill. We thank the three anonymous referees for comments and suggestions that have helped to improve the presentation of this paper.

References

  1. 1.
    Agdeppa, R.P., Yamashita, N., Fukushima, M.: Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem. Pac. J. Optim. 6(1), 3–19 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anitescu, M., Petra, C.: Higher-order confidence intervals for stochastic programming using bootstrapping. Technical Report ANL/MCS-P1964-1011, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL (2011)Google Scholar
  3. 3.
    Attouch, H., Cominetti, R., Teboulle, M.: Forward: Special issue on nonlinear convex optimization and variational inequalities. Math. Program. 116(1–2), 1–3 (2009).  https://doi.org/10.1007/s10107-007-0116-6 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, X., Pong, T.K., Wets, R.J.B.: Two-stage stochastic variational inequalities: an ERM-solution procedure (2015) (preprint)Google Scholar
  6. 6.
    Chen, X., Wets, R.J.B., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649–673 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dentcheva, D., Römisch, W.: Differential stability of two-stage stochastic programs. SIAM J. Optim. 11(1), 87–112 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6(4), 1087–1105 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dupacova, J., Wets, R.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Stat. 16(4), 1517–1549 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)zbMATHGoogle Scholar
  12. 12.
    Fang, H., Chen, X., Fukushima, M.: Stochastic R\(_0\) matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ferris, M.C., Pang, J.S.: Complementarity and Variational Problems: State of the Art. SIAM, Philadelphia (1997)zbMATHGoogle Scholar
  14. 14.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Genz, A., Bretz, F.: Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics. Springer, Heidelberg (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., Hothorn, T.: mvtnorm: multivariate normal and t distributions (2013). http://CRAN.R-project.org/package=mvtnorm. R package version 0.9-9996
  17. 17.
    Giannessi, F., Maugeri, A. (eds.): Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York (1995)zbMATHGoogle Scholar
  18. 18.
    Giannessi, F., Maugeri, A., Pardalos, P.M. (eds.): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Nonconvex optimization and its applications, vol. 58. Kluwer Academic Publishers, Dordrecht (2001)zbMATHGoogle Scholar
  19. 19.
    Gürkan, G., Pang, J.S.: Approximations of Nash equilibria. Math. Program. 117(1–2), 223–253 (2009).  https://doi.org/10.1007/10107-007-0156-y MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gürkan, G., Yonca Özge, A., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms, and applications. Math. Program. 48, 161–220 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Haurie, A., Zaccour, G., Legrand, J., Smeers, Y.: A stochastic dynamic nash-cournot model for the European gas market. Technical Report G-87-24, École des hautes études commerciales, Montréal, Québec, Canada (1987)Google Scholar
  23. 23.
    Huber, P.: The behavior of maximum likelihood estimates under nonstandard conditions. In: LeCam L., Neyman J. (eds.) Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics, pp. 221–233. University of California Press, Berkeley, CA (1967)Google Scholar
  24. 24.
    Jiang, H., Xu, H.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53(6), 1462–1475 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    King, A.J., Rockafellar, R.T.: Asymptotic theory for solutions in statistical estimation and stochastic programming. Math. Oper. Res. 18, 148–162 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lamm, M., Lu, S., Budhiraja, A.: Individual confidence intervals for true solutions to expected value formulations stochastic variational inequalities. Math. Prog. Ser. B 165(1), 151–196 (2017)CrossRefzbMATHGoogle Scholar
  27. 27.
    Lan, G., Nemirovski, A., Shapiro, A.: Validation analysis of mirror descent stochastic approximation method. Math. Program. 134(2), 425–458 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Linderoth, J., Shapiro, A., Wright, S.: The empirical behavior of sampling methods for stochastic programming. Ann. Oper. Res. 142, 215–241 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lu, S.: A new method to build confidence regions for solutions of stochastic variational inequalities. Optimization 63(9), 1431–1443 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lu, S.: Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities. SIAM J. Optim. 24(3), 1458–1484 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lu, S., Budhiraja, A.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38, 545–568 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lu, S., Liu, Y., Yin, L., Zhang, K.: Confidence intervals and retions for the lasso by using stochastic variational inequality techniques in optimization. J. R. Stat. Soc. Ser. B 79(2), 589–611 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Luo, M., Lin, G.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Pang, J.: Newton’s method for B-differentiable equations. Math. Oper. Res. 15, 311–341 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Pang, J.S., Ralph, D.: Forward: Special issue on nonlinear programming, variational inequalities, and stochastic programming. Math. Program. 117(1–2), 1–4 (2009).  https://doi.org/10.1007/s10107-007-0169-6 MathSciNetCrossRefGoogle Scholar
  36. 36.
    Phelps, C., Royset, J.O., Gong, Q.: Optimal control of uncertain systems using sample average approximations. SIAM J. Control Optim. 54(1), 1–29 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ralph, D.: On branching numbers of normal manifolds. Nonlinear Anal. Theory Methods Appl. 22, 1041–1050 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5(1), 43–62 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16(2), 292–309 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Robinson, S.M.: Normal maps induced by linear transformations. Math. Oper. Res. 17(3), 691–714 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Robinson, S.M.: Sensitivity analysis of variational inequalities by normal-map techniques. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 257–269. Plenum Press, New York (1995)CrossRefGoogle Scholar
  42. 42.
    Rockafellar, R.T., Wets, R.J.B.: Stochastic variational inequalities: single-stage to multistage. Math. Program. Ser. B 165(1), 331–360 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Römisch, W.: Stability of stochastic programming problems. In: Ruszczyński, A., Shapiro, A. (eds.) Handbooks in Operations Research and Management Science, vol. 10, pp. 483–554. Elsevier, Amsterdam (2003)Google Scholar
  44. 44.
    Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  45. 45.
    Shapiro, A.: Asymptotic behavior of optimal solutions in stochastic programming. Math. Oper. Res. 18, 829–845 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.P.: Lectures on Stochastic Programming: Modeling and Theory. Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia, PA (2009)Google Scholar
  47. 47.
    Shapiro, A., Homem-de Mello, T.: On the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs. SIAM J. Optim. 11(1), 70–86 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optimization 57, 395–418 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Stefanski, L.A., Boos, D.D.: The calculus of M-estimation. Am. Stat. 56(1), 29–38 (2002)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Vogel, S.: Universal confidence sets for solutions of optimization problems. SIAM J. Optim. 19(3), 1467–1488 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Wald, A.: Note on the consitency of the maximum likelihood estimate. Ann. Math. Stat. 20, 595–601 (1949)CrossRefzbMATHGoogle Scholar
  52. 52.
    Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia Pac. J. Oper. Res. 27(1), 103–119 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Yin, L., Lu, S., Liu, Y.: Confidence intervals for sparse penalized regression with random designs (2015) (submitted for publication)Google Scholar
  54. 54.
    Zhang, C., Chen, X., Sumlee, A.: Robust Wardrop’s user equilibrium assignment under stochastic demand and supply. Transp. Res. B 45(3), 534–552 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.SAS Institute IncCaryUSA
  2. 2.Department of Statistics and Operations ResearchUniversity of North Carolina at Chapel HillChapel HillUSA

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