Skip to main content
SpringerLink
Log in

Quasi-Monte Carlo methods for linear two-stage stochastic programming problems

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

Quasi-Monte Carlo (QMC) algorithms are studied for generating scenarios to solve two-stage linear stochastic programming problems. Their integrands are piecewise linear-quadratic, but do not belong to the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to \(L_{2}\). This implies that randomly shifted lattice rules may achieve the optimal rate of convergence \(O(n^{-1+\delta })\) with \(\delta \in (0,\frac{1}{2}]\) and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two. The geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution is normal. We discuss effective dimensions and techniques for dimension reduction. Numerical experiments for a production planning model with normal inputs show that indeed convergence rates close to the optimal rate are achieved when using randomly shifted lattice rules or scrambled Sobol’ point sets accompanied with principal component analysis for dimension reduction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bratley, P., Fox, B.L.: Algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 14, 88–100 (1988)

    Article  MATH  Google Scholar 

  2. Brockwell, P.J., Davis, R.A.: Introduction to Time Series and Forecasting, 2nd edn. Springer, New York (2002)

    Book  MATH  Google Scholar 

  3. Dick, J.: Walsh spaces containing smooth functions and Quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the Quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  6. Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Liberating the weights. J. Complex. 20, 593–623 (2004)

    Article  MATH  Google Scholar 

  7. Drew, S.S., Homem-de-Mello, T.: Quasi-Monte Carlo strategies for stochastic optimization. In: Proceedings of the 2006 Winter Simulation Conference, IEEE pp. 774–782 (2006)

  8. Eichhorn, A., Römisch, W., Wegner, I.: Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. In: IEEE St. Petersburg Power Tech (2005)

  9. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  10. Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18, 209–232 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Glasserman, P.: Monte-Carlo Methods in Financial Engineering. Springer, New York (2003)

    Book  Google Scholar 

  12. Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complex. 26, 455–489 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Griebel, M., Kuo, F.Y., Sloan, I.H.: The smoothing effect of the ANOVA decomposition. J. Complex. 26, 523–551 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Griebel, M., Kuo, F.Y., Sloan, I.H.: The smoothing effect of integration in \({\mathbb{R}}^{d}\) and the ANOVA decomposition. Math. Comput. 82, 383–400 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Heitsch, H., Leövey, H., Römisch, W.: Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs? Stochastic Programming E-Print Series 5 (2012) (http://www.speps.org) and submitted

  16. Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67, 299–322 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19, 293–325 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  18. Homem-de-Mello, T.: On rates of convergence for stochastic optimization problems under non-i.i.d. sampling. SIAM J. Optim. 19, 524–551 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Joe, S., Kuo, F.Y.: Remark on algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 29, 49–57 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Koivu, M.: Variance reduction in sample approximations of stochastic programs. Math. Program. 103, 463–485 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence in weighted Korobov and Sobolev spaces. J. Complex. 19, 301–320 (2003)

    Article  MATH  Google Scholar 

  22. Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond. ANZIAM J. 53, 1–37 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kuo, F.Y., Sloan, I.H.: Lifting the curse of dimensionality. Notices Am. Math. Soc. 52, 1320–1328 (2005)

    MATH  MathSciNet  Google Scholar 

  24. Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Waterhouse, B.J.: Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands. J. Complex. 26, 135–160 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Woźniakowski, H.: On decomposition of multivariate functions. Math. Comput. 79, 953–966 (2010)

    Article  MATH  Google Scholar 

  26. L’Ecuyer, P., Lemieux, Ch.: Recent advances in randomized quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovski, F. (eds.). Modeling Uncertainty, pp. 419–474. Kluwer, Boston (2002)

  27. Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer, New York (2009)

    MATH  Google Scholar 

  28. Liu, R., Owen, A.B.: Estimating mean dimensionality of analysis of variance decompositions. J. Am. Stat. Assoc. 101, 712–721 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Matoušek, J.: On the \(L_{2}\)-discrepancy for anchored boxes. J. Complex. 14, 527–556 (1998)

    Article  MATH  Google Scholar 

  30. Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8, 3–30 (1998)

    Article  MATH  Google Scholar 

  31. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  32. Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nožička, F., Guddat, J., Hollatz, H., Bank, B.: Theory of Linear Parametric Programming (in German). Akademie-Verlag, Berlin (1974)

    Google Scholar 

  34. Nuyens, D., Cools, R.: Fast algorithms for component-by-component constructions of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75, 903–922 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Owen, A.B.: Randomly permuted \((t, m, s)\)-nets and \((t, s)\)-sequences. In: Niederreiter, H., Shiue, P.J.S. (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics, vol. 106, pp. 299–317. Springer, New York (1995)

  36. Owen, A.B.: Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34, 1884–1910 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Owen, A.B.: The dimension distribution and quadrature test functions. Stat. Sinica 13, 1–17 (2003)

    MATH  Google Scholar 

  38. Owen, A.B.: Multidimensional variation for Quasi-Monte Carlo. In: Fan, J., Li, G. (eds.) International Conference on Statistics, pp. 49–74. World Scientific, Singapore (2005)

    Google Scholar 

  39. Owen, A.B.: Better estimation of small Sobol’ sensitivity indices. ACM Trans. Model. Comput. Simul. 23, 11:1–11:17 (2013)

    Article  Google Scholar 

  40. Papageorgiou, A.: Brownian bridge does not offer a consistent advantage in Quasi-Monte Carlo integration. J. Complex. 18, 171–186 (2002)

    Article  MATH  Google Scholar 

  41. Pennanen, T., Koivu, M.: Epi-convergent discretizations of stochastic programs via integration quadratures. Numer. Math. 100, 141–163 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  43. Römisch, W.: Stability of stochastic programming problems. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10, pp. 483–554. Elsevier, Amsterdam (2003)

    Google Scholar 

  44. Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10. Elsevier, Amsterdam (2003)

    Google Scholar 

  45. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, MPS-SIAM Series on Optimization, Philadelphia, 2009 (2014 Second Edition)

  46. Sloan, I.H., Woźniakowski, H.: When are Quasi Monte Carlo algorithms efficient for high-dimensional integration. J. Complex. 14, 1–33 (1998)

    Article  MATH  Google Scholar 

  47. Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40, 1650–1665 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  48. Sobol’, I.M.: The distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7, 86–112 (1967)

    Article  MathSciNet  Google Scholar 

  49. Sobol’, I.M.: Multidimensional Quadrature Formulas and Haar Functions. Nauka, Moscow (1969). (in Russian)

    MATH  Google Scholar 

  50. Sobol’, I.M.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55, 271–280 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  51. Sobol’, I.M., Asotsky, D., Kreinin, A., Kucherenko, S.: Construction and comparison of high-dimensional Sobol’ generators. Wilmott 56, 64–79 (2011)

    Article  Google Scholar 

  52. Takemura, A.: Tensor analysis of ANOVA decomposition. J. Am. Stat. Assoc. 78, 894–900 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  53. Triebel, H.: Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration. European Mathematical Society, Zürich (2010)

  54. Walkup, D., Wets, R.J.-B.: Lifting projections of convex polyedra. Pac. J. Math. 28, 465–475 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  55. Wang, X., Fang, K.-T.: The effective dimension and Quasi-Monte Carlo integration. J. Complex. 19, 101–124 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  56. Wang, X., Sloan, I.H.: Why are high-dimensional finance problems often of low effective dimension. SIAM J. Sci. Comput. 27, 159–183 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  57. Wang, X., Sloan, I.H.: Quasi-Monte Carlo methods in financial engineering: an equivalence principle and dimension reduction. Oper. Res. 59, 80–95 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  58. Wets, R.J.-B.: Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Rev. 16, 309–339 (1974)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The research of the first author was supported by grant GR 705/13 of the German Research Foundation (DFG) and the second by the DFG Research Center Matheon at Berlin. The authors wish to thank all referees for their helpful comments and valuable suggestions that improved readability and presentation.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Humboldt-University Berlin, Berlin, Germany

    H. Leövey & W. Römisch

Authors
  1. H. Leövey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Römisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. Römisch.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leövey, H., Römisch, W. Quasi-Monte Carlo methods for linear two-stage stochastic programming problems. Math. Program. 151, 315–345 (2015). https://doi.org/10.1007/s10107-015-0898-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0898-x

Mathematics Subject Classification

Search

Navigation