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.
Similar content being viewed by others
References
Baldeaux, J.: Higher order nets and sequences, PhD Thesis, The University of New South Wales, Sydney (2010)
Brockwell, P.J., Davis, R.A.: Introduction to Time Series and Forecasting, 2nd edn. Springer, New York (2002)
Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Liberating the weights. J. Complex. 20, 593–623 (2004)
Dick, J.: Walsh spaces containing smooth functions and Quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 1519–1553 (2008)
Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge (2010)
Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration—the Quasi-Monte Carlo way. Acta Numerica 22, 133–288 (2013)
Drew, S.S., Homem-de-Mello, T.: Quasi-Monte Carlo strategies for stochastic optimization. Proceedings of the 2006 Winter Simulation Conference. IEEE, Piscataway (2006)
Dudley, R.M.: The speed of mean Glivenko–Cantelli convergence. Ann. Math. Stat. 40, 40–50 (1969)
Eichhorn, A., Römisch, W., Wegner, I.: Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. IEEE St. Petersburg Power Tech (2005)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Glasserman, P.: Monte-Carlo Methods in Financial Engineering. Springer, New York (2003)
Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000)
Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complex. 26, 455–489 (2010)
Griebel, M., Kuo, F.Y., Sloan, I.H.: The smoothing effect of the ANOVA decomposition. J. Complex. 26, 523–551 (2010)
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)
Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67, 299–322 (1998)
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)
Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19, 293–325 (1948)
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)
Hong, H.S., Hickernell, F.J.: Algorithm 823: Implementing scrambled digital sequences. ACM Trans. Math. Softw. 29, 95–109 (2003)
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)
Joe, S., Kuo, F.Y.: Remark on Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 29, 49–57 (2003)
Koivu, M.: Variance reduction in sample approximations of stochastic programs. Math. Program. 103, 463–485 (2005)
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)
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)
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)
Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Woźniakowski, H.: On decomposition of multivariate functions. Math. Comput. 79, 953–966 (2010)
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)
Lemieux, Ch.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer, New York (2009)
Liu, R., Owen, A.B.: Estimating mean dimensionality of analysis of variance decompositions. J. Am. Stat. Assoc. 101, 712–721 (2006)
Matoušek, J.: On the \(L_{2}\)-discrepancy for anchored boxes. J. Complex. 14, 527–556 (1998)
Matsumoto, M., Nishimura, T.: Mersenne Twister: A 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8, 3–30 (1998)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Nožička, F., Guddat, J., Hollatz, H., Bank, B.: Theory of Linear Parametric Programming (in German). Akademie-Verlag, Berlin (1974)
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)
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)
Owen, A.B.: Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34, 1884–1910 (1997)
Owen, A.B.: Scrambled net variance for integrals of smooth functions. Ann. Stat. 25, 1541–1562 (1997)
Owen, A.B.: The dimension distribution and quadrature test functions. Statistica Sinica 13, 1–17 (2003)
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)
Owen, A.B.: Local antithetic sampling with scrambled nets. Ann. Stat. 36, 2319–2343 (2008)
Pagès, G.: A space quantization method for numerical integration. J. Comput. Appl. Math. 89, 1–38 (1997)
Papageorgiou, A.: Brownian bridge does not offer a consistent advantage in Quasi-Monte Carlo integration. J. Complex. 18, 171–186 (2002)
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)
Pennanen, T., Koivu, M.: Epi-convergent discretizations of stochastic programs via integration quadratures. Numerische Mathematik 100, 141–163 (2005)
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)
Proinov, P.D.: Discrepancy and integration of continuous functions. J. Approx. Theory 52, 121–131 (1998)
Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. I. Springer, New York (1998)
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)
Römisch, W.: Scenario generation. In: Cochran, J.J. (ed.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)
Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10. Elsevier, Amsterdam (2003)
Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming. MPS-SIAM Series on Optimization, Philadelphia (2009)
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)
Sloan, I.H., Woźniakowski, H.: When are Quasi Monte Carlo algorithms efficient for high-dimensional integration. J. Complex. 14, 1–33 (1998)
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)
Sobol’, I.M.: Multidimensional Quadrature Formulas and Haar Functions (in Russian). Nauka, Moscow (1969)
Sobol’, I.M.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55, 271–280 (2001)
Sobol’, I.M., Kucherenko, S.: Derivative based global sensitivity measures and their link with global sensitivity indices. Math. Comput. Simul. 79, 3009–3017 (2009)
Takemura, A.: Tensor analysis of ANOVA decomposition. J. Am. Stat. Assoc. 78, 894–900 (1983)
Tezuka, Shu, Faure, Henri: I-binomial scrambling of digital nets and sequences. J. Complex. 19, 744–757 (2003)
Walkup, D., Wets, R.J.-B.: Lifting projections of convex polyedra. Pac. J. Math. 28, 465–475 (1969)
Wallace, S.W., Ziemba, W.T. (eds.): Applications of Stochastic Programming. MPS-SIAM Series on Optimization, Philadelphia (2005)
Wang, X.: Tractability of multivariate integration using Quasi-Monte Carlo algorithms. Math. Comput. 72, 823–838 (2003)
Wang, X.: On the effects of dimension reduction techniques on some high-dimensional problems in finance. Oper. Res. 54, 1063–1078 (2006)
Wang, X., Fang, K.-T.: The effective dimension and Quasi-Monte Carlo integration. J. Complex. 19, 101–124 (2003)
Wang, X., Sloan, I.H.: Why are high-dimensional finance problems often of low effective dimension. SIAM J. Sci. Comput. 27, 159–183 (2005)
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)
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)
Wang, X., Sloan, I.H.: Quasi-Monte Carlo methods in financial engineering: An equivalence principle and dimension reduction. Oper. Res. 59, 80–95 (2011)
Wets, R.J.-B.: Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Rev. 16, 309–339 (1974)
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
Corresponding author
Rights and permissions
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-016-9843-z