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The impact of sampling methods on bias and variance in stochastic linear programs

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Abstract

Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample average approximation (SAA). The value of the objective function at the optimal solution obtained via SAA provides an estimate of the true optimal objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates (AV) and Latin Hypercube (LH) sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. For our analytic example and the eight test problems we derive or estimate the condition number as defined in Shapiro et al. (Math. Program. 94:1–19, 2002). We find that for ill-conditioned problems, bias plays a larger role in MSE, and AV and LH sampling methods are more likely to reduce bias.

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References

  1. Avramidis, A., Wilson, J.: Correlation-induction techniques for estimating quantiles in simulation experiments. Oper. Res. 46, 574–591 (1998)

    Article  MATH  Google Scholar 

  2. Bailey, T.G., Jensen, P., Morton, D.P.: Response surface analysis of two-stage stochastic linear programming with recourse. Nav. Res. Logist. 46, 753–778 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beale, E.M.: On minimizing a convex function subject to linear inequalities. J. R. Stat. Soc. B 17, 173–184 (1955)

    MATH  MathSciNet  Google Scholar 

  4. Burnetas, A., Smith, C.: Adaptive ordering and pricing for perishable products. Oper. Res. 48, 436–443 (2000)

    Article  Google Scholar 

  5. Czyzyk, J., Linderoth, J., Shen, J.: SUTIL: A utility library for handling stochastic programs. user’s manual (2005). Available at http://coral.ie.lehigh.edu/sutil

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Diwekar, U.M., Kalagnanam, J.R.: Efficient sampling technique for optimization under uncertainty. AIChE J. 43, 440–447 (1997)

    Article  Google Scholar 

  9. Dupačová, J., Wets, R.J.-B.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Stat. 16, 1517–1549 (1988)

    Article  MATH  Google Scholar 

  10. Foster, I., Kesselman, C.: Computational grids. In: Foster, I., Kesselman, C. (eds.) The Grid: Blueprint for a New Computing Infrastructure, pp. 15–52. Morgan Kaufmann, San Mateo (1999)

    Google Scholar 

  11. Higle, J.L.: Variance reduction and objective function evaluation in stochastic linear programs. INFORMS J. Comput. 10, 236–247 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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 

  13. Infanger, G.: Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs. Ann. Oper. Res. 39, 69–95 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Janjarassuk, U., Linderoth, J.: Reformulation and sampling to solve a stochastic network interdiction problem. Networks 52, 120–132 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Law, A., Kelton, D.: Simulation Modeling and Analysis, 3rd edn. McGraw-Hill, Boston (2000)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Linderoth, J., Wright, S.: Implementing a decomposition algorithm for stochastic programming on a computational grid. Comput. Optim. Appl. 24, 207–250 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Litzkow, M.J., Livny, M., Mutka, M.W.: Condor—A hunter of idle workstations. In: Proceedings of the 8th International Conference on Distributed Computing Systems, pp. 104–111 (1988)

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

    Chapter  Google Scholar 

  20. Mak, W.K., Morton, D.P., Wood, R.K.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24, 47–56 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mulvey, J.M., Ruszczyński, A.: A new scenario decomposition method for large scale stochastic optimization. Oper. Res. 43, 477–490 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Norkin, V., Pflug, G., Ruszczyński, A.: A branch and bound method for stochastic global optimization. Math. Program. 83, 425–450 (1998)

    MATH  Google Scholar 

  24. Partani, A.: Adaptive jackknife estimators for stochastic programming. Ph.D. thesis, University of Texas at Austin (2007)

  25. Partani, A., Morton, D., Popova, I.: Jackknife estimators for reducing bias in asset allocation. In: Perrone, L.F., Wieland, F.P., Liu, J., Lawson, B.G., Nicol, D.M., Fujimoto, R.M. (eds.) Proceedings of the 2006 Winter Simulation Conference, pp. 783–791. IEEE Press, New York (2006)

    Chapter  Google Scholar 

  26. Powell, W., Topaloglu, H.: Fleet management. In: Wallace, S., Ziemba, W. (eds.) Applications of Stochastic Programming. MPS-SIAM Series on Optimization, pp. 185–216. SIAM, Philadelphia (2005)

    Chapter  Google Scholar 

  27. Sen, S., Doverspike, R.D., Cosares, S.: Network planning with random demand. Telecommun. Syst. 3, 11–30 (1994)

    Article  Google Scholar 

  28. Shapiro, A., Homem-de-Mello, T., Kim, J.: Conditioning of convex piecewise linear stochastic programs. Math. Program. 94, 1–19 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Van Slyke, R., Wets, R.J.-B.: L-shaped linear programs with applications to control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Douglas J. Thomas.

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Freimer, M.B., Linderoth, J.T. & Thomas, D.J. The impact of sampling methods on bias and variance in stochastic linear programs. Comput Optim Appl 51, 51–75 (2012). https://doi.org/10.1007/s10589-010-9322-x

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