Random Search Under Additive Noise

  • Luc Devroye
  • Adam Krzyzak
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 46)


Simulated Annealing Modeling Uncertainty Additive Noise Random Search Stochastic Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aarts, E. and J. Korst. (1989). Simulated Annealing and Boltzmann Machines, John Wiley, New York.zbMATHGoogle Scholar
  2. Ackley, D.H. (1987). A Connectionist Machine for Genetic Hillclimbing, Kluwer Academic Publishers, Boston.Google Scholar
  3. Aluffi-Pentini, F., V. Parisi, and F. Zirilli. (1985). “Global optimization and stochastic differential equations,” Journal of Optimization Theory and Applications, vol. 47, pp. 1–16.MathSciNetzbMATHGoogle Scholar
  4. Anily, S. and A. Federgruen. (1987). “Simulated annealing methods with general acceptance probabilities,” Journal of Applied Probability, vol. 24, pp. 657–667.MathSciNetzbMATHGoogle Scholar
  5. Banzhaf, W., P. Nordin, and R. E. Keller. (1988). Genetic Programming: An Introduction: On the Automatic Evolution of Computer Programs and Its Applications, Morgan Kaufman, San Mateo, CA.zbMATHGoogle Scholar
  6. Baum, L. and M. Katz. (1965). “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 121, pp. 108–123.MathSciNetzbMATHGoogle Scholar
  7. Becker, R.W. and G. V. Lago. (1970). “A global optimization algorithm,” in: Proceedings of the 8th Annual Allerton Conference on Circuit and System Theory, pp. 3–12.Google Scholar
  8. Bekey, G.A. and M. T. Ung. (1974). “A comparative evaluation of two global search algorithms,” IEEE Transactions on Systems, Man and Cybernetics, vol. SMC-4, pp. 112–116.zbMATHGoogle Scholar
  9. G. L. Bilbro, G.L. and W. E. Snyder. (1991). “Optimization of functions with many minima,” IEEE Transactions on Systems, Man and Cybernetics, vol. SMC-21, pp. 840–849.MathSciNetGoogle Scholar
  10. Boender, C.G.E., A. H. G. Rinnooy Kan, L. Stougie, and G. T. Timmer. (1982). “A stochastic method for global optimization,” Mathematical Programming, vol. 22, pp. 125–140.MathSciNetzbMATHGoogle Scholar
  11. Bohachevsky, I.O. (1986). M. E. Johnson, and M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics, vol. 28, pp. 209–217.zbMATHGoogle Scholar
  12. Bremermann, H.J. (1962). “Optimization through evolution and recombination,” in: Self-Organizing Systems, (edited by M. C. Yovits, G. T. Jacobi and G. D. Goldstein), pp. 93–106, Spartan Books, Washington, D.C.Google Scholar
  13. Bremermann, H.J. (1968). “Numerical optimization procedures derived from biological evolution processes,” in: Cybernetic Problems in Bionics, (edited by H. L. Oestreicher and D. R. Moore), pp. 597–616, Gordon and Breach Science Publishers, New York.Google Scholar
  14. Brooks, S.H. (1958). “A discussion of random methods for seeking maxima,” Operations Research, vol. 6, pp. 244–251.Google Scholar
  15. Brooks, S.H. (1959). “A comparison of maximum-seeking methods,” Operations Research, vol. 7, pp. 430–457.Google Scholar
  16. Bäck, T., F. Hoffmeister, and H.-P. Schwefel. (1991). “A survey of evolution strategies,” in: Proceedings of the Fourth International Conference on Ge-netic Algorithms, (edited by R. K. Belew and L. B. Booker), pp. 2–9, Morgan Kaufman Publishers, San Mateo, CA.Google Scholar
  17. Cerny, V. (1985). “Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm,” Journal of Optimization Theory and Applications, vol. 45, pp. 41–51.MathSciNetzbMATHGoogle Scholar
  18. Dekkers, A. and E. Aarts. (1991). “Global optimization and simulated annealing,” Mathematical Programming, vol. 50, pp. 367–393.MathSciNetzbMATHGoogle Scholar
  19. Devroye, L. (1972). “The compound random search algorithm,” in: Proceedings of the International Symposium on Systems Engineering and Analysis, Purdue University, vol. 2, pp. 195–110.Google Scholar
  20. Devroye, L. (1976). “On the convergence of statistical search,” IEEE Transactions on Systems, Man and Cybernetics, vol. SMC-6, pp. 46–56.MathSciNetzbMATHGoogle Scholar
  21. Devroye, L. (1976). “On random search with a learning memory,” in: Proceedings of the IEEE Conference on Cybernetics and Society, Washington, pp. 704–711.Google Scholar
  22. L. Devroye, L. (1977). “An expanding automaton for use in stochastic optimization,” Journal of Cybernetics and Information Science, vol. 1, pp. 82–94.Google Scholar
  23. Devroye, L. (1978a). “The uniform convergence of nearest neighbor regression function estimators and their application in optimization,” IEEE Transactions on Information Theory, vol. IT-24, pp. 142–151.MathSciNetzbMATHGoogle Scholar
  24. Devroye, L. (1978b). “Rank statistics in multimodal stochastic optimization,” Technical Report, School of Computer Science, McGill University.Google Scholar
  25. Devroye, L. (1978c). “Progressive global random search of continuous functions,” Mathematical Programming, vol. 15, pp. 330–342.MathSciNetzbMATHGoogle Scholar
  26. Devroye, L. (1979). “Global random search in stochastic optimization problems,” in: Proceedings of Optimization Days 1979, Montreal.Google Scholar
  27. de Biase, L. and F. Frontini. (1978). “A stochastic method for global optimization: its structure and numerical performance,” in: Towards Global Optimization 2, (edited by L. C. W. Dixon and G. P. Szegö pp. 85–102, North Holland, Amsterdam.zbMATHGoogle Scholar
  28. Dvoretzky, A., J. C. Kiefer, and J. Wolfowitz. (1956). “Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator,” Annals of Mathematical Statistics, vol. 27, pp. 642–669.MathSciNetzbMATHGoogle Scholar
  29. Ermakov, S.M. and A. A. Zhiglyavskii. (1983). “On random search for a global extremum,” Theory of Probability and its Applications, vol. 28, pp. 136–141.MathSciNetGoogle Scholar
  30. Ermoliev, Yu. and R. Wets. (1988). “Stochastic programming, and introduction,” in: Numerical Techniques of Stochastic Optimization, (edited by R. J.-B. Wets and Yu. M. Ermoliev), pp. 1–32, Springer-Verlag, New York.zbMATHGoogle Scholar
  31. Fisher, L. and S. J. Yakowitz. (1976). “Uniform convergence of the potential function algorithm,” SIAM Journal on Control and Optimization, vol. 14, pp. 95–103.MathSciNetzbMATHGoogle Scholar
  32. Gastwirth, J.L. (1966). “On robust procedures,” Journal of the American Statistical Association, vol. 61, pp. 929–948.MathSciNetzbMATHGoogle Scholar
  33. Gaviano, M. (1975). “Some general results on the convergence of random search algorithms in minimization problems,” in: Towards Global Optimization, (edited by L. C. W. Dixon and G. P. Szegö), pp. 149–157, North Holland, New York.Google Scholar
  34. Geffroy, J. (1958). “Contributions à la théorie des valeurs extrêmes,” Publications de l’Institut de Statistique des Universités de Paris, vol. 7, pp. 37–185.MathSciNetGoogle Scholar
  35. Gelfand, S.B. and S. K. Mitter. (1991). “Weak convergence of Markov chain sampling methods and annealing algorithms to diffusions,” Journal of Optimization Theory and Applications, vol. 68, pp. 483–498.MathSciNetzbMATHGoogle Scholar
  36. Geman, S. and C.-R. Hwang. (1986). “Diffusions for global optimization,” SIAM Journal on Control and Optimization, vol. 24, pp. 1031–1043.MathSciNetzbMATHGoogle Scholar
  37. Gidas, B. (1985). “Global optimization via the Langevin equation,” in: Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, pp. 774–778.Google Scholar
  38. Gnedenko, A.B.V. (1943). Sur la distribution du terme maximum d’une série aléatoire, Annals of Mathematics, vol. 44, pp. 423–453.MathSciNetGoogle Scholar
  39. Goldberg, D.E. (1989). Genetic Algorithms in Search Optimization and Machine Learning, Addison-Wesley, Reading, Mass.zbMATHGoogle Scholar
  40. Gurin, L.S. (1966). “Random search in the presence of noise,” Engineering Cybernetics, vol. 4, pp. 252–260.MathSciNetGoogle Scholar
  41. Gurin, L.S. and L. A. Rastrigin. (1965). “Convergence of the random search method in the presence of noise,” Automation and Remote Control, vol. 26, pp. 1505–1511.zbMATHGoogle Scholar
  42. Haario, H. and E. Saksman. (1991). “Simulated annealing process in general state space,” Advances in Applied Probability, vol. 23, pp. 866–893.MathSciNetzbMATHGoogle Scholar
  43. Hajek, B. (1988). “Cooling schedules for optimal annealing,” Mathematics of Operations Research, vol. 13, pp. 311–329.MathSciNetzbMATHGoogle Scholar
  44. Hajek, B. and G. Sasaki. (1989). “Simulated annealing—to cool or not,” Systems and Control Letters, vol. 12, pp. 443–447.MathSciNetzbMATHGoogle Scholar
  45. Holland, J.H. (1973). “Genetic algorithms and the optimal allocation of trials,” SIAM Journal on Computing, vol. 2, pp. 88–105.MathSciNetzbMATHGoogle Scholar
  46. Holland, J.H. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis With Applications to Biology, Control, and Artificial Intelligence, MIT Press, Cambridge, Mass.Google Scholar
  47. Jarvis, R.A. (1975). “Adaptive global search by the process of competitive evolution,” IEEE Transactions on Systems, Man and Cybernetics, vol. SMC-5, pp. 297–311.zbMATHGoogle Scholar
  48. Johnson, D.S., C. R. Aragon, L. A. McGeogh, and C. Schevon. (1989). “Optimization by simulated annealing: an experimental evaluation; part I, graph partitioning,” Operations Research, vol. 37, pp. 865–892.zbMATHGoogle Scholar
  49. Rinnooy Kan, A.H.G. and G. T. Timmer. (1984). “Stochastic methods for global optimization,” American Journal of Mathematical and Management Sciences, vol. 4, pp. 7–40.MathSciNetzbMATHGoogle Scholar
  50. Karmanov, V.G. (1974). “Convergence estimates for iterative minimization methods,” USSR Computational Mathematics and Mathematical Physics, vol. 14(1), pp. 1–13.MathSciNetzbMATHGoogle Scholar
  51. Kiefer, J. and J. Wolfowitz. (1952). “Stochastic estimation of the maximum of a regression function,” Annals of Mathematical Statistics, vol. 23, pp. 462–466.MathSciNetzbMATHGoogle Scholar
  52. Kirkpatrick, S., C. D. Gelatt, and M. P. Vecchi. (1983). “Optimization by simulated annealing,” Science, vol. 220, pp. 671–680.MathSciNetzbMATHGoogle Scholar
  53. Koronacki, J. (1976). “Convergence of random-search algorithms,” Automatic Control and Computer Sciences, vol. 10(4), pp. 39–45.MathSciNetGoogle Scholar
  54. Kushner, H.L. (1987). “Asymptotic global behavior for stochastic approximation via diffusion with slowly decreasing noise effects: global minimization via Monte Carlo,” SIAM Journal on Applied Mathematics, vol. 47, pp. 169–185.MathSciNetzbMATHGoogle Scholar
  55. Lai, T.L. and H. Robbins. (1985) “Asymptotically efficient adaptive allocation rules,” Advances in Applied Mathematics, vol. 6, pp. 4–22.MathSciNetzbMATHGoogle Scholar
  56. Mann, H.B. and D. R. Whitney. (1947). “On a test of whether one or two random variables is stochastically larger than the other,” Annals of Mathematical Statistics, vol. 18, pp. 50–60.MathSciNetzbMATHGoogle Scholar
  57. Marti, K. (1982). “Minimizing noisy objective functions by random search methods,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 62, pp. T377–T380.MathSciNetzbMATHGoogle Scholar
  58. Marti, K. (1992). “Stochastic optimization in structural design,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 72, pp. T452–T464.MathSciNetzbMATHGoogle Scholar
  59. Massart, P. (1990). “The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality,” Annals of Probability, vol. 18, pp. 1269–1283.MathSciNetzbMATHGoogle Scholar
  60. Matyas, J. (1965). “Random optimization,” Automation and Remote Control, vol. 26, pp. 244–251.MathSciNetzbMATHGoogle Scholar
  61. Meerkov, S.M. (1972). “Deceleration in the search for the global extremum of a function,” Automation and Remote Control, vol. 33, pp. 2029–2037.zbMATHGoogle Scholar
  62. Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. (1953). “Equation of state calculation by fast computing machines,” Journal of Chemical Physics, vol. 21, pp. 1087–1092.Google Scholar
  63. Mockus, J.B. (1989). Bayesian Approach to Global Optimization, Kluwer Academic Publishers, Dordrecht, Netherlands.zbMATHGoogle Scholar
  64. Männer, R. and H.-P. Schwefel. (1991). “Parallel Problem Solving from Nature,” vol. 496, Lecture Notes in Computer Science, Springer-Verlag, Berlin.zbMATHGoogle Scholar
  65. Petrov, V.V. (1975). Sums of Independent Random Variables, Springer-Verlag, Berlin.zbMATHGoogle Scholar
  66. Pinsky, M.A. (1991). Lecture Notes on Random Evolution, World Scientific Publishing Company, Singapore.Google Scholar
  67. Pintér, J. (1984). “Convergence properties of stochastic optimization procedures,” Mathematische Operationsforschung und Statistik, Series Optimization, vol. 15, pp. 405–427.MathSciNetzbMATHGoogle Scholar
  68. Pintér, J. (1996). Global Optimization in Action, Kluwer Academic Publishers, Dordrecht.zbMATHGoogle Scholar
  69. Price, W.L. (1983). “Global optimization by controlled random search,” Journal of Optimization Theory and Applications, vol. 40, pp. 333–348.MathSciNetzbMATHGoogle Scholar
  70. Rechenberg, I. (1973). Evolutionsstrategie—Optimierung technischer Systeme nach Prinzipien der biologischen Evolution, Frommann-Holzboog, Stuttgart.Google Scholar
  71. Rinnooy Kan, A.H.G. and G. T. Timmer. (1987). “Stochastic global optimization methods part II: multi level methods,” Mathematical Programming, vol. 39, pp. 57–78.MathSciNetzbMATHGoogle Scholar
  72. Rinnooy Kan, A.H.G. and G. T. Timmer. (1987). “Stochastic global optimization methods part I: clustering methods,” Mathematical Programming, vol. 39, pp. 27–56.MathSciNetzbMATHGoogle Scholar
  73. Robbins, H. (1952). “Some aspects of the sequential design of experiments,” Bulletin of the American Mathematical Society, vol. 58, pp. 527–535.MathSciNetzbMATHGoogle Scholar
  74. Rubinstein, R. Y. and I. Weissman. (1979). “The Monte Carlo method for global optimization,” Cahiers du Centre d’Etude de Recherche Operationelle, vol. 21, pp. 143–149.MathSciNetzbMATHGoogle Scholar
  75. Schumer, M.A. and K. Steiglitz. (1968). “Adaptive step size random search,” IEEE Transactions on Automatic Control, vol. AC-13, pp. 270–276.Google Scholar
  76. Schwefel, H.-P. (1977). Modellen mittels der Evolutionsstrategie, Birkhäuser Verlag, Basel.zbMATHGoogle Scholar
  77. Schwefel, H.-P. (1981). Numerical Optimization of Computer Models, John Wiley, Chichester.zbMATHGoogle Scholar
  78. Schwefel, H.-P. (1995). Evolution and Optimum Seeking, Wiley, New York.zbMATHGoogle Scholar
  79. Sechen, C. (1988). VLSI Placement and Global Routing using Simulated Annealing, Kluwer Academic Publishers.Google Scholar
  80. Shorack, G.R. and J. A. Wellner. (1986). Empirical Processes with Applications to Statistics, John Wiley, New York.zbMATHGoogle Scholar
  81. Shubert, B.O. (1972). “A sequential method seeking the global maximum of a function,” SIAM Journal on Numerical Analysis, vol. 9, pp. 379–388.MathSciNetzbMATHGoogle Scholar
  82. F. J. Solis, F.J. and R. B. Wets. (1981). “Minimization by random search techniques,” Mathematics of Operations Research, vol. 1, pp. 19–30.MathSciNetzbMATHGoogle Scholar
  83. Tarasenko, G.S. (1977). “Convergence of adaptive algorithms of random search,” Cybernetics, vol. 13, pp. 725–728.Google Scholar
  84. Törn, A. (1974). Global Optimization as a Combination of Global and Local Search, Skriftserie Utgiven av Handelshogskolan vid Abo Akademi, Abo, Finland.Google Scholar
  85. Törn, A. (1976). “Probabilistic global optimization, a cluster analysis approach,” in: Proceedings of the EURO II Conference, Stockholm, Sweden, pp. 521–527, North Holland, Amsterdam.Google Scholar
  86. Törn, A. and A. Žilinskas. (1989). Global Optimization, Lecture Notes in Computer Science, vol. 350, Springer-Verlag, Berlin.zbMATHGoogle Scholar
  87. Uosaki, K., H. Imamura, M. Tasaka, and H. Sugiyama. (1970). “A heuristic method for maxima searching in case of multimodal surfaces,” Technology Reports of Osaka University, vol. 20, pp. 337–344.Google Scholar
  88. Vanderbilt, D. and S. G. Louie. (1984). “A Monte Carlo simulated annealing approach to optimization over continuous variables,” Journal of Computational Physics, vol. 56, pp. 259–271.MathSciNetzbMATHGoogle Scholar
  89. Van Laarhoven, P.J.M. and E. H. L. Aarts. (1987). Simulated Annealing: Theory and Applications, D. Reidel, Dordrecht.zbMATHGoogle Scholar
  90. Wasan, M.T. (1969). Stochastic Approximation, Cambridge University Press, New York.zbMATHGoogle Scholar
  91. Wilcoxon, F. (1945). “Individual comparisons by ranking methods,” Biometrics Bulletin, vol. 1, pp. 80–83.Google Scholar
  92. Yakowitz, S. (1992). “Automatic learning: theorems for concurrent simulation and optimization,” in: 1992 Winter Simulation Conference Proceedings, (edited by J. J. Swain, D. Goldsman, R. C. Crain and J. R. Wilson), pp. 487–493, ACM, Baltimore, MD.Google Scholar
  93. Yakowitz, S.J. (1989). “A statistical foundation for machine learning, with application to go-moku,” Computers and Mathematics with Applications, vol. 17, pp. 1095–1102.MathSciNetzbMATHGoogle Scholar
  94. Yakowitz, S.J. (1989). “A globally-convergent stochastic approximation,” Technical Report, Systems and Industrial Engineering Department, University of Arizona, Tucson, AZ.zbMATHGoogle Scholar
  95. Yakowitz, S.J. (1989). “On stochastic approximation and its generalizations,” Technical Report, Systems and Industrial Engineering Department, University of Arizona, Tucson, AZ, 1989.zbMATHGoogle Scholar
  96. Yakowitz, S.J. (1992). “A decision model and methodology for the AIDS epidemic,” Applied Mathematics and Computation, vol. 55, pp. 149–172.MathSciNetzbMATHGoogle Scholar
  97. Yakowitz, S.J. (1993). “Global stochastic approximation,” SIAM Journal on Control and Optimization, vol. 31, pp. 30–40.MathSciNetzbMATHGoogle Scholar
  98. Yakowitz, S.J. and L. Fisher. (1973). “On sequential search for the maximum of an unknown function,” Journal of Mathematical Analysis and Applications, vol. 41, pp. 234–259.MathSciNetzbMATHGoogle Scholar
  99. Yakowitz, S.J., R. Hayes, and J. Gani. (1992). “Automatic learning for dynamic Markov fields, with applications to epidemiology,” Operations Research, vol. 40, pp. 867–876.zbMATHGoogle Scholar
  100. Yakowitz, S.J., T. Jayawardena, and S. Li. (1992). “Theory for automatic learning under Markov-dependent noise, with applications,” IEEE Transactions on Automatic Control, vol. AC-37, pp. 1316–1324.zbMATHGoogle Scholar
  101. Yakowitz, S.J. and M. Kollier. (1992). “Machine learning for blackjack counting strategies,” Journal of Forecasting and Statistical Planning, vol. 33, pp. 295–309.MathSciNetzbMATHGoogle Scholar
  102. Yakowitz, S.J. and W. Lowe. (1991). “Nonparametric bandit methods,” Annals of Operations Research, vol. 28, pp. 297–312.MathSciNetzbMATHGoogle Scholar
  103. Yakowitz, S.J. and E. Lugosi. (1990). “Random search in the presence of noise, with application to machine learning,” SIAM Journal on Scientific and Statistical Computing, vol. 11, pp. 702–712.MathSciNetzbMATHGoogle Scholar
  104. Yakowitz, S.J. and A. Vesterdahl. (1993). “Contribution to automatic learning with application to self-tuning communication channel,” Technical Report, Systems and Industrial Engineering Department, University of Arizona.Google Scholar
  105. Zhigljavsky, A.A. (1991). Theory of Global Random Search, Kluwer Academic Publishers, Hingham, MA.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2002

Authors and Affiliations

  • Luc Devroye
    • 1
  • Adam Krzyzak
    • 2
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Department of Computer ScienceConcordia UniversityMontrealCanada

Personalised recommendations