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Letter to the Editor—A Monte Carlo Method for the Approximate Solution of Certain Types of Constrained Optimization Problems

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  • Martin Pincus

    (Polytechnic Institute of Brooklyn, Brooklyn, New York)

Abstract

This paper considers the problem of minimizing a function F ( x 1 , …, x n ) over a closed, bounded region S in n -dimensional space under the assumption that there exists a unique minimizing point ( z 1 , …, z n )ϵ S . In a previous paper I represented the coordinates of the minimizing point as the limit of a ratio of integrals. The same type of ratio appears, in a different context, in statistical mechanics where a Monte Carlo method has been developed, by Metropolis et al., for its numerical evaluation. The purpose of this paper is to point out the connection of Metropolis's method with the above type of minimisation problem. The idea of the method is to associate with the minimization problem a Markov chain whose sample averages converge with probability one to (approximately) the minimizing point ( z 1 , …, z n ). The Markov chain should be easily realizable on a computer. An estimate of the error from sampling over a finite time period is given.

Suggested Citation

  • Martin Pincus, 1970. "Letter to the Editor—A Monte Carlo Method for the Approximate Solution of Certain Types of Constrained Optimization Problems," Operations Research, INFORMS, vol. 18(6), pages 1225-1228, December.
    • Handle: RePEc:inm:oropre:v:18:y:1970:i:6:p:1225-1228
    DOI: 10.1287/opre.18.6.1225
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    Cited by:

    1. Tahir Ekin & Stephen Walker & Paul Damien, 2023. "Augmented simulation methods for discrete stochastic optimization with recourse," Annals of Operations Research, Springer, vol. 320(2), pages 771-793, January.
    2. Tsionas, Mike G., 2023. "Joint production in stochastic non-parametric envelopment of data with firm-specific directions," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1336-1347.
    3. Shahid Khan & Mohammed M. A. Almazah & Ataur Rahman & Ijaz Hussain, 2023. "Optimization of Meteorological Monitoring Network of New South Wales, Australia," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 37(9), pages 3395-3419, July.
    4. Genetha Anne Gray & Tamara G. Kolda & Ken Sale & Malin M. Young, 2004. "Optimizing an Empirical Scoring Function for Transmembrane Protein Structure Determination," INFORMS Journal on Computing, INFORMS, vol. 16(4), pages 406-418, November.
    5. Alejandro Estrada-Moreno & Albert Ferrer & Angel A. Juan & Javier Panadero & Adil Bagirov, 2020. "The Non-Smooth and Bi-Objective Team Orienteering Problem with Soft Constraints," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    6. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci, 2019. "A Characterization of Probabilities with Full Support and the Laplace Method," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 470-478, May.
    7. Tsionas, Mike G., 2020. "A coherent approach to Bayesian Data Envelopment Analysis," European Journal of Operational Research, Elsevier, vol. 281(2), pages 439-448.
    8. Tahir Ekin & Nicholas G. Polson & Refik Soyer, 2014. "Augmented Markov Chain Monte Carlo Simulation for Two-Stage Stochastic Programs with Recourse," Decision Analysis, INFORMS, vol. 11(4), pages 250-264, December.
    9. Yanzhi Li & Andrew Lim & Brian Rodrigues, 2005. "Manpower allocation with time windows and job‐teaming constraints," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(4), pages 302-311, June.
    10. Zubarev, A.Yu. & Iskakova, L.Yu., 1996. "Statistical thermodynamics of ferronematic," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 229(2), pages 203-217.
    11. Tahir Ekin & Nicholas G. Polson & Refik Soyer, 2017. "Augmented nested sampling for stochastic programs with recourse and endogenous uncertainty," Naval Research Logistics (NRL), John Wiley & Sons, vol. 64(8), pages 613-627, December.
    12. L. Ingber, 2020. "Forecasting with importance-sampling and path-integrals: Applications to COVID-19," Lester Ingber Papers 20fi, Lester Ingber.
    13. Roy, Dilip Kumar & Lal, Alvin & Sarker, Khokan Kumer & Saha, Kowshik Kumar & Datta, Bithin, 2021. "Optimization algorithms as training approaches for prediction of reference evapotranspiration using adaptive neuro fuzzy inference system," Agricultural Water Management, Elsevier, vol. 255(C).
    14. Ekin, Tahir, 2018. "Integrated maintenance and production planning with endogenous uncertain yield," Reliability Engineering and System Safety, Elsevier, vol. 179(C), pages 52-61.
    15. Tevfik Aktekin & Tahir Ekin, 2016. "Stochastic call center staffing with uncertain arrival, service and abandonment rates: A Bayesian perspective," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(6), pages 460-478, September.
    16. Zubarev, A.Yu. & Iskakova, L.Yu., 1996. "Structure transformations in ferrosmectics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 224(3), pages 489-502.

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