Variance Reduction in Population-Based Optimization: Application to Unit Commitment

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9554)


We consider noisy optimization and some traditional variance reduction techniques aimed at improving the convergence rate, namely (i) common random numbers (CRN), which is relevant for population-based noisy optimization and (ii) stratified sampling, which is relevant for most noisy optimization problems. We present artificial models of noise for which common random numbers are very efficient, and artificial models of noise for which common random numbers are detrimental. We then experiment on a desperately expensive unit commitment problem. As expected, stratified sampling is never detrimental. Nonetheless, in practice, common random numbers provided, by far, most of the improvement.


Noisy optimization Variance reduction Stratified sampling Common random numbers 


  1. 1.
    Bellman, R.: Dynamic Programming. Princeton Univ, Press (1957)zbMATHGoogle Scholar
  2. 2.
    Billingsley, P.: Probability and Measure. John Wiley and Sons, New York (1986)zbMATHGoogle Scholar
  3. 3.
    Cranley, R., Patterson, T.: Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13(6), 904–914 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Defourny, B.: Machine learning solution methods for multistage stochastic programming. Ph.D. thesis, Institut Montefiore, Université de Liège (2010)Google Scholar
  5. 5.
    Dowell, M., Jarratt, P.: The “pegasus” method for computing the root of an equation. BIT Numer. Math. 12(4), 503–508 (1972). Google Scholar
  6. 6.
    Dupacov, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming: an approach using probability metrics. Math. Programm. 95, 3 (2003). No. 20 in Stochastic Programming E-Print Series, Institut fr Mathematik. Springer, Berlin (2000). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hamzaçebi, C., Kutay, F.: Continuous functions minimization by dynamic random search technique. Appl. Math. Model. 31(10), 2189–2198 (2007)CrossRefzbMATHGoogle Scholar
  8. 8.
    Huang, S.-C., Coulom, R., Lin, S.-S.: Monte-carlo simulation balancing in practice. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2010. LNCS, vol. 6515, pp. 81–92. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Kleinman, N.L., Spall, J.C., Naiman, D.Q.: Simulation-based optimization with stochastic approximation using common random numbers. Manag. Sci. 45(11), 1570–1578 (1999). CrossRefzbMATHGoogle Scholar
  10. 10.
    Kozak, M.: Optimal stratification using random search method in agricultural surveys. Stat. Trans. 6(5), 797–806 (2004).
  11. 11.
    Lavallée, P., Hidiroglou, M.: On the stratification of skewed populations. Surv. Method. 14(1), 33–43 (1988).
  12. 12.
    Linderoth, J., Shapiro, A., Wright, S.: The empirical behavior of sampling methods for stochastic programming. Ann. OR 142(1), 215–241 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mascagni, M., Chi, H.: On the scrambled halton sequence. Monte-Carlo Methods Appl. 10(3), 435–442 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    de Matos, V., Philpott, A., Finardi, E.: Improving the performance of stochastic dual dynamic programming. Applications - OR and Management Sciences (Scheduling) (2012).
  15. 15.
    Morokoff, W.J.: Generating quasi-random paths for stochastic processes. 40(4), 765–788.
  16. 16.
    Niederreiter, H.: Random number generation and quasi-monte carlo methods (1992)Google Scholar
  17. 17.
    Pereira, M.V.F., Pinto, L.: Multi-stage stochastic optimization applied to energy planning. Math. Program. 52(2), 359–375. Google Scholar
  18. 18.
    Sethi, V.: A note on optimum stratification of populations for estimating the population means. Aust. J. Stat. 5(1), 20–33 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shapiro, A., Tekaya, W., da Costa, J.P., Soares, M.P.: Risk neutral and risk averse stochastic dual dynamic programming method. Eur. J. Oper. Res. 224(2), 375–391 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Storn, R., Price, K.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Strens, M., Lx, H.G., Moore, A., Brodley, E., Danyluk, A.: Policy search using paired comparisons. J. Mach. Learn. Res. 3, 921–950 (2002)Google Scholar
  22. 22.
    Strens, M., Moore, A.: Direct policy search using paired statistical tests. In: Proceedings of the 18th International Conference on Machine Learning, pp. 545–552. Morgan Kaufmann, San Francisco (2001)Google Scholar
  23. 23.
    Takagi, H., Pallez, D.: Paired comparison-based interactive differential evolution. In: World Congress on Nature & Biologically Inspired Computing, NaBIC 2009, pp. 475–480. IEEE (2009)Google Scholar
  24. 24.
    Wang, X., Hickernell, F.: Randomized halton sequences. Math. Comput. Model. 32, 887–899 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zabinsky, Z.B.: Random Search Algorithms: Encyclopedia of Operations Research and Management Science. Wiley, New York (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Univ. Paris-SudGif-sur-YvetteFrance

Personalised recommendations