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One-dimensional Global Optimization Based on Statistical Models

  • James M. Calvin
  • Antanas Žilinskas
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 59)

Abstract

This paper presents a review of global optimization methods based on statistical models of multimodal functions. The theoretical and methodological aspects are emphasized.

Keywords

Optimization statistical models convergence 

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Bibliography

  1. [1]
    Kushner, H.: A versatile stochastic model of a function of unknown and time-varying form, J. Math. Anal. Appl. 5 (1962), 150–167.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [2]
    Mockus, J.: On Bayesian methods of search for extremum, Automatics and Computers 3 (1972), 53–62 (in Russian).MathSciNetzbMATHGoogle Scholar
  3. [3]
    Mockus, J.: Bayesian Approach to Global Optimization, Kluwer Acad. Publ., Dordrecht, 1989.zbMATHGoogle Scholar
  4. [4]
    Boender, G. and Romeijn, E.: Stochastic methods, In: R. Horst and P. Pardalos (eds), Handbook of Global Optimization, Kluwer Acad. Publ., Dordrecht, 1995, pp. 829–869.Google Scholar
  5. [5]
    Törn, A. and Žilinskas, A.: Global Optimization, Springer, 1989.Google Scholar
  6. [6]
    Strongin, R.: Numerical Methods in Multiextremal Optimization (in Russian), Nauka, 1978.Google Scholar
  7. [7]
    Horst, R. and Tuy, H.: Global Optimization — Deterministic Approaches, 2nd edn, Springer, 1992.Google Scholar
  8. [8]
    Zhigljavski, A. and Žilinskas, A.: Methods of Search for Global Extremum (in Russian), Nauka, Moscow, 1991.Google Scholar
  9. [9]
    Kushner, H.: A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise, J. Basic Eng. 86 (1964), 97–106.Google Scholar
  10. [10]
    Žilinskas A.: On global one-dimensional optimization, Engrg. Cybernet., Izv. AN USSR 4 (1976), 71–74 (in Russian).Google Scholar
  11. [11]
    Archetti, F. and Betro, B.: A probabilistic algorithm for global optimization, Calcolo 16 (1979), 335–343.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Žilinskas, A.: Optimization of one-dimensional multimodal functions, algorithm 133. Appl. Statist. 23 (1978), 367–385.Google Scholar
  13. [13]
    Žilinskas, A.: Mimun-optimization of one-dimensional multimodal functions in the presence of noise, algoritmus 44, Apl. Mat. 25 (1980), 392–402.Google Scholar
  14. [14]
    Žilinskas A.: Two algorithms for one-dimensional multimodal minimization, Mathematische Operationsforschung und Statistik, ser. Optimization 12 (1981), 53–63.zbMATHGoogle Scholar
  15. [15]
    Locatelli, M.: Baeysian algorithms for one-dimensional global optimization, J. Global Optim. 10 (1997), 57–76.CrossRefMathSciNetzbMATHGoogle Scholar
  16. [16]
    Calvin, J. M. and Glynn, P. W.: Average case behavior of random search for the maximum, J. Appl. Probab. 34 (1997), 631–642.MathSciNetGoogle Scholar
  17. [17]
    Ritter, K.: Approximation and optimization on the Wiener space, J. Complexity 6 (1990), 337–364.CrossRefMathSciNetzbMATHGoogle Scholar
  18. [18]
    Shepp, L. A.: The joint density of the maximum and its location for a Wiener process with drift, J. Appl. Probab. 16 (1976), 423–427.MathSciNetGoogle Scholar
  19. [19]
    Lindgren, G.: Local maxima of Gaussian fields, Ark. Math. 10 (1972), 195–218.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Calvin, J. M. and Žilinskas, A.: On convergence of a p-algorithm based on a statistical model of continuously differentiable functions, J. Global Optim. 19 (2001), 229–245.CrossRefMathSciNetzbMATHGoogle Scholar
  21. [21]
    Calvin, J. M. and Žilinskas, A.: On convergence of the p-algorithm for one-dimensional global optimization of smooth functions, J. Optim. Theory Appl. 102(3) (1999), 479–495.CrossRefMathSciNetzbMATHGoogle Scholar
  22. [22]
    Neimark, J. and Strongin, R.: Information approach to search for minimum of a function, Izv. AN SSSR, Engineering Cybernetics No. 1 (1966), 17–26 (in Russian).Google Scholar
  23. [23]
    Fine, T.: Optimal search for location of the maximum of a unimodal function, IEEE Trans. Inform. Theory (2) (1966), 103–111.Google Scholar
  24. [24]
    Converse, A.: The use of uncertainty in a simultaneos search, Oper. Res. 10 (1967), 1088–1095.Google Scholar
  25. [25]
    Heyman, M.: Optimal simultaneos search for the maximum by the principle of statistical information, Oper. Res. (1968), 1194–1205.Google Scholar
  26. [26]
    Neuman, P.: An asymptotically optimal procedure for searching a zero or an extremum of a function, In: Proceedings of 2nd Prague Symp. Asymp. Statist., 1981, pp. 291–302.Google Scholar
  27. [27]
    Timonov, L.: An algorithm for search of a global extremum, Engrg. Cybernet. 15 (1977), 38–44.MathSciNetGoogle Scholar
  28. [28]
    Shaltenis, V.: Structure Analysis of Optimization Problems (in Russian), Mokslas, Vilnius, 1989.Google Scholar
  29. [29]
    Žilinskas, A.: On statistical models for multimodal optimization, Mathematische Operationsforschung und Statistik, ser. Statistics 9 (1978), 255–266.zbMATHGoogle Scholar
  30. [30]
    Žilinskas, A.: Axiomatic approach to statistical models and their use in multimodal optimization theory, Math. Programming 22 (1982), 104–116.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Žilinskas, A.: A reviewo f statistical models for global optimization, J. Global Optim. (1992), 145–153.Google Scholar
  32. [32]
    Luce, D. and Suppes, P.: Preference, utility and subjective probability, In: R. Bush, D. Luce and E. Galanter (eds), Handbook of Global Optimization, Wiley, 1965, pp. 249–410.Google Scholar
  33. [33]
    Žilinskas, A.: Axiomatic approach to extrapolation problem under uncertainty, Automatics and Remote Control, No. 12 (1979), 66–70 (in Russian).Google Scholar
  34. [34]
    Gutman, H.-M.: A radial basis function method for global optimization, J. Global Optim. 19 (2001), 201–227.MathSciNetGoogle Scholar
  35. [35]
    Žilinskas, A.: Axiomatic characterization of a global optimization algorithm and investigation of its search strategies, Oper. Res. Lett. 4 (1985), 35–39.MathSciNetzbMATHGoogle Scholar
  36. [36]
    Shagen, I.: Internal modelling of objective functions for global optimization, J. Optim. Theory Appl. 51 (1986), 345–353.MathSciNetGoogle Scholar
  37. [37]
    Žilinskas A.: Statistical models for global optimization by means of select and clone, Optimization 48 (2000), 117–135.MathSciNetzbMATHGoogle Scholar
  38. [38]
    Žilinskas, A.: One-step Bayesian method for the search of the optimium of one-variable functions, Cybernetics, No. 1 (1975), 139–144 (in Russian).Google Scholar
  39. [39]
    Calvin, J.M. and Žilinskas, A.: On the choice of statistical model for one-dimensional p-algorithm, Control and Cybernet. 29(2) (2000), 555–565.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Calvin, J. M.: Convergence rate of the p-algorithm for optimization of continuous functions, In: P. Pardalos (ed.), Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, Kluwer Acad. Publ., Boston, 1999, pp. 116–129.Google Scholar
  41. [41]
    Calvin, J.M. and Žilinskas, A.: A one-dimensional p-algorithm with convergence rate o(n −3+δ) for smooth functions, J. Optim. Theory Appl. 106 (2000), 297–307.CrossRefMathSciNetzbMATHGoogle Scholar
  42. [42]
    Törn, A. and Žilinskas, A.: Parallel global optimization algorithms in optimal design, Lecture Notes in Control and Inform. Sci. 143, Springer, 1990, pp. 951–960.Google Scholar
  43. [43]
    Locatelli, M. and Schoen, F.: An adaptive stochastic global optimization algorithm for one-dimensional functions, Ann. Oper. Res. 58 (1995), 263–278.CrossRefMathSciNetzbMATHGoogle Scholar
  44. [44]
    Sergeyev, Y.: An information global optimization algorithm with local tuning, SIAM J. Optim. 5 (1995), 858–870.CrossRefMathSciNetzbMATHGoogle Scholar
  45. [45]
    Žilinskas, A.: Note on Pinter’s paper, Optimization 19 (1988), 195.MathSciNetGoogle Scholar
  46. [46]
    Žilinskas A.: A note on extended univariate algorithms by J. Pinter, Computing 41 (1989), 275–276.MathSciNetzbMATHGoogle Scholar
  47. [47]
    Gergel, V. and Sergeev, Y.: Sequential and parallel algorithms for global minimizing functions with Lipshitz derivatives, Internat. J. Comput. Math. Appl. 37 (1999), 163–179.zbMATHGoogle Scholar
  48. [48]
    Sergeyev, Y.: Parallel information algorithm with local tuning for solving multidimensional global optimization problems, J. Global Optim. 15 (1999), 157–167.CrossRefMathSciNetzbMATHGoogle Scholar
  49. [49]
    Hansen, P. and Jaumard, B.: Lipshitz optimization, In: R. Horst and P. Pardalos (eds), Handbook of Global Optimization, Kluwer Acad. Publ., Dordrecht, 1995, pp. 404–493.Google Scholar
  50. [50]
    Pinter, J.: Extended univariate algorithms for n-dimensional global optimization, Computing 36 (1986), 91–103.MathSciNetzbMATHGoogle Scholar
  51. [51]
    Hansen, P. and Jaumard, B.: On Timonov’s algorithm for global optimization of univariate Lipshitz functions, J. Global Optim. 1 (1991), 37–46.MathSciNetzbMATHGoogle Scholar
  52. [52]
    Al-Mharmah, H. and Calvin, J. M.: Optimal random non-adaptive algorithm for global optimization of brownian motion, J. Global Optim. 8 (1996), 81–90.CrossRefMathSciNetzbMATHGoogle Scholar
  53. [53]
    Ritter, K.: Average-Case Analysis of Numerical Problems, Lecture Notes in Math. 1733, 2000.Google Scholar
  54. [54]
    Calvin, J. M.: Average performance of passive algorithms for global optimization, J. Math. Anal. Appl. 191 (1995), 608–617.CrossRefMathSciNetzbMATHGoogle Scholar
  55. [55]
    Calvin, J. M. and Glynn, P. M.: Complexity of non-adaptive optimization algorithms for a class of diffusions. Comm. Statist. Stochastic Models 12 (1996), 343–365.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    Novak, E.: Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Math. 1349, Springer, 1988.Google Scholar
  57. [57]
    Calvin, J. M.: A one-dimensional optimization algorithm and its convergence rate under the wiener measure, J. Complexity (2001), accepted.Google Scholar
  58. [58]
    Calvin, J. M.: Average performance of a class of adaptive algorithms for global optimization, Ann. Appl. Probab. 7 (1997), 711–730.MathSciNetzbMATHGoogle Scholar
  59. [59]
    Törn, A., Viitanen, S. and Ali, M.: Stochastic global optimization: Problem classes and solution techniques, J. Global Optim. 14 (1999), 437–447.MathSciNetzbMATHGoogle Scholar
  60. [60]
    Floudas, C. and Pardalos, P.: Handbook of Test Problems in Local and Global Optimization, Kluwer Acad. Publ.. Dordrecht, 1999.zbMATHGoogle Scholar
  61. [61]
    Wingo, D.: Fitting three parameter lognormal model by numerical global optimization—an improved algorithm. Computational Statistics and Data Analysis, No. 2 (1984), 13–25.Google Scholar
  62. [62]
    Orsier, B. and Pellegrini, C.: Using global line searches for finding global minima of mlp error functions. In: International Conference on Neural Networks and their Applications, Marseilles, 1997, pp. 229–235.Google Scholar
  63. [63]
    Groch, A., Vidigal, L. and Director, S.: A new global optimization method for electronic circuit design, IEEE Trans. on Circuits and Systems 32 (1985), 160–170.CrossRefGoogle Scholar
  64. [64]
    Perttunen, C., Jones, D. and Stuckman, B.: Lipshitzian optimization without the Lipshitz constant, J. Optim. Theory Appl. 79 (1993), 157–181.MathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • James M. Calvin
    • 1
  • Antanas Žilinskas
    • 2
  1. 1.Department of Computer and Information ScienceNew Jersey Institute of TechnologyNewarkUSA
  2. 2.Institute of Mathematics and InformaticsVMUVilniusLithuania

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