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Nelder-Mead Simplex Modifications for Simulation Optimization

Author

Listed:
  • Russell R. Barton

    (Department of Industrial and Systems Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802)

  • John S. Ivey, Jr.

    (The Eastman Kodak Company, Rochester, New York 14650)

Abstract

When the Nelder-Mead method is used to optimize the expected response of a stochastic system (e.g., an output of a discrete-event simulation model), the simplex-resizing steps of the method introduce risks of inappropriate termination. We give analytical and empirical results describing the performance of Nelder-Mead when it is applied to a response function that incorporates an additive white-noise error, and we use these results to develop new modifications of Nelder-Mead that yield improved estimates of the optimal expected response. Compared to Nelder-Mead, the best performance was obtained by a modified method, RS + S9, in which (a) the best point in the simplex is reevaluated at each shrink, step and (b) the simplex is reduced by 10% (rather than 50%) at each shrink step. In a suite of 18 test problems that were adapted from the MINPACK collection of NETLIB, the expected response at the estimated optimal point obtained by RS + S9 had errors that averaged 15% less than at the original method's estimated optimal point, at an average cost of three times as many function evaluations. Two well-known existing modifications for stochastic responses, the (n + 3)-rule and the next-to-worst rule, were found to be inferior to the new modification RS + S9.

Suggested Citation

  • Russell R. Barton & John S. Ivey, Jr., 1996. "Nelder-Mead Simplex Modifications for Simulation Optimization," Management Science, INFORMS, vol. 42(7), pages 954-973, July.
    • Handle: RePEc:inm:ormnsc:v:42:y:1996:i:7:p:954-973
    DOI: 10.1287/mnsc.42.7.954
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    Cited by:

    1. Fan, Shu-Kai S. & Zahara, Erwie, 2007. "A hybrid simplex search and particle swarm optimization for unconstrained optimization," European Journal of Operational Research, Elsevier, vol. 181(2), pages 527-548, September.
    2. Hachicha, Wafik & Ammeri, Ahmed & Masmoudi, Faouzi & Chachoub, Habib, 2010. "A comprehensive literature classification of simulation optimisation methods," MPRA Paper 27652, University Library of Munich, Germany.
    3. Kuo-Hao Chang & L. Jeff Hong & Hong Wan, 2013. "Stochastic Trust-Region Response-Surface Method (STRONG)---A New Response-Surface Framework for Simulation Optimization," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 230-243, May.
    4. Chang, Kuo-Hao, 2015. "A direct search method for unconstrained quantile-based simulation optimization," European Journal of Operational Research, Elsevier, vol. 246(2), pages 487-495.
    5. Chang, Kuo-Hao, 2012. "Stochastic Nelder–Mead simplex method – A new globally convergent direct search method for simulation optimization," European Journal of Operational Research, Elsevier, vol. 220(3), pages 684-694.
    6. Kao, Chiang & Chen, Shih-Pin, 2006. "A stochastic quasi-Newton method for simulation response optimization," European Journal of Operational Research, Elsevier, vol. 173(1), pages 30-46, August.
    7. David J. Eckman & Shane G. Henderson & Sara Shashaani, 2023. "Diagnostic Tools for Evaluating and Comparing Simulation-Optimization Algorithms," INFORMS Journal on Computing, INFORMS, vol. 35(2), pages 350-367, March.
    8. Rosen, Scott L. & Harmonosky, Catherine M. & Traband, Mark T., 2007. "A simulation optimization method that considers uncertainty and multiple performance measures," European Journal of Operational Research, Elsevier, vol. 181(1), pages 315-330, August.
    9. Cheung, Ka Chun & He, Wanting & Wang, He, 2023. "Multi-constrained optimal reinsurance model from the duality perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 199-214.
    10. Neddermeijer, H.G. & Piersma, N. & van Oortmarssen, G.J. & Habbema, J.D.F. & Dekker, R., 1999. "Comparison of response surface methodology and the Nelder and Mead simplex method for optimization in microsimulation models," Econometric Institute Research Papers EI 9924-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    11. Arsham H., 1998. "Techniques for Monte Carlo Optimizing," Monte Carlo Methods and Applications, De Gruyter, vol. 4(3), pages 181-230, December.
    12. W. Liu & Y. H. Dai, 2001. "Minimization Algorithms Based on Supervisor and Searcher Cooperation," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 359-379, November.
    13. Pinto, Roberto, 2016. "Stock rationing under a profit satisficing objective," Omega, Elsevier, vol. 65(C), pages 55-68.
    14. Galip Altinay, 2003. "Estimating growth rate in the presence of serially correlated errors," Applied Economics Letters, Taylor & Francis Journals, vol. 10(15), pages 967-970.
    15. M Laguna & J Molina & F Pérez & R Caballero & A G Hernández-Díaz, 2010. "The challenge of optimizing expensive black boxes: a scatter search/rough set theory approach," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(1), pages 53-67, January.
    16. David G. Humphrey & James R. Wilson, 2000. "A Revised Simplex Search Procedure for Stochastic Simulation Response Surface Optimization," INFORMS Journal on Computing, INFORMS, vol. 12(4), pages 272-283, November.
    17. Jeroen J. van den Broek & Nicolien T. van Ravesteyn & Eveline A. Heijnsdijk & Harry J. de Koning, 2018. "Simulating the Impact of Risk-Based Screening and Treatment on Breast Cancer Outcomes with MISCAN-Fadia," Medical Decision Making, , vol. 38(1_suppl), pages 54-65, April.
    18. Alkhamis, Talal M. & Ahmed, Mohamed A., 2006. "A modified Hooke and Jeeves algorithm with likelihood ratio performance extrapolation for simulation optimization," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1802-1815, November.
    19. Ozden, Mufit & Ho, Yu-Chi, 2003. "A probabilistic solution-generator for simulation," European Journal of Operational Research, Elsevier, vol. 146(1), pages 35-51, April.
    20. Sudarshan Kumar & Tiziana Di Matteo & Anindya S. Chakrabarti, 2020. "Disentangling shock diffusion on complex networks: Identification through graph planarity," Papers 2001.01518, arXiv.org.
    21. Satyajith Amaran & Nikolaos V. Sahinidis & Bikram Sharda & Scott J. Bury, 2016. "Simulation optimization: a review of algorithms and applications," Annals of Operations Research, Springer, vol. 240(1), pages 351-380, May.

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