Skip to main content
SpringerLink
Log in

Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in 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 that first and second order ANOVA terms have mixed first order partial derivatives and belong to \(L_{2}\). Hence, randomly shifted lattice rules (SLR) 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 at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point sets accompanied with PCA 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
Fig. 4

Similar content being viewed by others

References

  1. Baldeaux, J.: Higher order nets and sequences, PhD Thesis, The University of New South Wales, Sydney (2010)

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  8. Dudley, R.M.: The speed of mean Glivenko–Cantelli convergence. Ann. Math. Stat. 40, 40–50 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Hickernell, F.J.: Obtaining \(O(N^{-2 +\epsilon })\) convergence for lattice quadrature rules. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 274–289. Springer, Berlin (2002)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. 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  MathSciNet  MATH  Google Scholar 

  20. Hong, H.S., Hickernell, F.J.: Algorithm 823: Implementing scrambled digital sequences. ACM Trans. Math. Softw. 29, 95–109 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Imai, J., Tan, K.S.: Minimizing effective dimension using linear transformation. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 275–292. Springer, Berlin (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. 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  MathSciNet  MATH  Google Scholar 

  25. 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  MathSciNet  MATH  Google Scholar 

  26. 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  MathSciNet  MATH  Google Scholar 

  27. 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 

  28. 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)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. 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 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

  35. 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  MathSciNet  MATH  Google Scholar 

  36. 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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Owen, A.B.: Scrambled net variance for integrals of smooth functions. Ann. Stat. 25, 1541–1562 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  41. Owen, A.B.: Local antithetic sampling with scrambled nets. Ann. Stat. 36, 2319–2343 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pagès, G.: A space quantization method for numerical integration. J. Comput. Appl. Math. 89, 1–38 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Pappas, SSp, Ekonomou, L., Karampelas, P., Karamousantas, D.C., Katsikas, S.K., Chatzarakis, G.E., Skafidas, P.D.: Electricity demand load forecasting of the Hellenic power system using an ARMA model. Electr. Power Syst. Res. 80, 256–264 (2010)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Pflug, G.Ch., Pichler, A.: Approximations for probability distributions and stochastic optimization problems. In: Bertocchi, M.I., Consigli, G. (eds.) Stochastic Optimization Methods in Finance and Energy, pp. 343–387. Springer, Berlin (2011)

  47. Proinov, P.D.: Discrepancy and integration of continuous functions. J. Approx. Theory 52, 121–131 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. I. Springer, New York (1998)

    MATH  Google Scholar 

  49. 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 

  50. Römisch, W.: Scenario generation. In: Cochran, J.J. (ed.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)

    Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

  53. Sloan, I.H.: QMC integration—beating intractability by weighting the coordinate directions. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 103–123. Springer, Berlin (2002)

    Chapter  Google Scholar 

  54. 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 

  55. 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  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Sobol’, I.M., Kucherenko, S.: Derivative based global sensitivity measures and their link with global sensitivity indices. Math. Comput. Simul. 79, 3009–3017 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  60. Tezuka, Shu, Faure, Henri: I-binomial scrambling of digital nets and sequences. J. Complex. 19, 744–757 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  62. Wallace, S.W., Ziemba, W.T. (eds.): Applications of Stochastic Programming. MPS-SIAM Series on Optimization, Philadelphia (2005)

    MATH  Google Scholar 

  63. Wang, X.: Tractability of multivariate integration using Quasi-Monte Carlo algorithms. Math. Comput. 72, 823–838 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  64. Wang, X.: On the effects of dimension reduction techniques on some high-dimensional problems in finance. Oper. Res. 54, 1063–1078 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  66. 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  MathSciNet  MATH  Google Scholar 

  67. Wang, X., Sloan, I.H.: Brownian bridge and principal component analysis: towards removing the curse of dimensionality. IMA J. Numer. Anal. 27, 631–654 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wang, X., Sloan, I.H.: Low discrepancy sequences in high dimensions: how well are their projections distributed? J. Comput. Appl. Math. 213, 366–386 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  69. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors wish to express their gratitude to Prof. Ian Sloan (University of New South Wales, Sydney) for inspiring conversations during his visit of the Humboldt-University Berlin in 2011. Much of the work on this paper was done during the first author held a position at the Humboldt-University Berlin. The research of the first author is partially supported by a grant of Kisters AG and by the Deutsche Forschungsgemeinschaft within SFB Transregio 154: Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks. The research of the second author is supported by a grant of the German Bundesministerium für Wirtschaft und Technologie (BMWi) and the third by the DFG Research Center Matheon at Berlin. The authors extend their gratitude to two anonymus referees and to the Associate Editor for their constructive and stimulating criticism.

Author information

Authors and Affiliations

  1. Weierstrass Institute, Berlin, Germany

    H. Heitsch

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

    H. Leövey & W. Römisch

Authors
  1. H. Heitsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Leövey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. 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

Heitsch, H., Leövey, H. & Römisch, W. Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?. Comput Optim Appl 65, 567–603 (2016). https://doi.org/10.1007/s10589-016-9843-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-016-9843-z

Keywords

Search

Navigation