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Axiomatic approach to statistical models and their use in multimodal optimization theory

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Abstract

This paper summarizes the results of axiomatic constructing statistical models of complicated multimodal functions. It is shown that an optimization algorithm may be constructed on the basis of a statistical model and some ideas of the rational choice theory. A brief review of related algorithms and reports on investigations of their efficiency is given.

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References

  1. P.E. Gill and W. Murrey, eds.,Numerical methods for constrained optimization (Academic Press, New York, 1974).

    Google Scholar 

  2. J. Mockus, “O bayesovych metodach poiska ekstremuma”,Avtomatika in Vyčislitelnaja Tehnika (Riga) 3 (1972) 53–62.

    Google Scholar 

  3. J. Mockus, “On Bayesian methods of seeking the extremum and their applications”, in: B. Gilchrist, ed.,Information Processing 77 (North-Holland, Amsterdam, 1977) pp. 195–200.

    Google Scholar 

  4. L. Savage, “Foundations of statistics reconsidered”, in: K. Borch and J. Mossin, eds.,Risk and uncertainty (MacMillan, New York, 1968) pp. 174–188.

    Google Scholar 

  5. A. Žilinskas, “On statistical models for multimodal optimization”,Mathematische Operationsforschung und Statistik, Series Statistics 9 (1978) 255–266.

    Google Scholar 

  6. L. Savage,Foundations of statistics (Wiley, New York, 1954).

    Google Scholar 

  7. M. De Groot,Optimal statistical decisions (McGraw-Hill, New York, 1970).

    Google Scholar 

  8. T. Fine,Theories of probability (Academic Press, New York, 1973).

    Google Scholar 

  9. H. Gottinger, “Konstruktion der subjektiven Wahrscheinlichkeiten”,Mathematische Operationsforschung und Statistik 5 (1974) 509–539.

    Google Scholar 

  10. McCrimon, “Descriptive and normative implications of the decision theory postulates”, in: K. Borch and J. Mossin, eds.,Risk and uncertainty (St. Martin, New York, 1968) pp. 3–23.

    Google Scholar 

  11. A. Žilinskas and A. Katkauskaite, “Postrojenije statisticheskich modelej slozhnych funkcij, vkluchajushchich elementy neopredelionosti”, in:VII Vsesojuznaja konferencija po teoriji kodirovanija i peredachi informaciji, Doklady, Chastj 1 (Vilnius, Moscow, 1978) pp. 70–74.

    Google Scholar 

  12. A. Žilinskas, “Aksiomaticheskij podchodk probleme ekstrapoliaciji v uslovijach neopredelionosti,Avtomatika i Telemehanika 12 (1979) 66–70.

    Google Scholar 

  13. T. Fine, “Extrapolation when very little is known about the source”,Information and Control 16 (1970) 331–359.

    Google Scholar 

  14. J. Goldman, “An approach to estimation and extrapolation with possible applications in an incompletely specified environment”,Information and Control 30 (1976) 203–223.

    Google Scholar 

  15. A. Žilinskas, “Issledovanije zadach ekstrapoliaciji v uslovijach neopredelionosti”, Teorija Optimal'nyh Rešenii (Vilnius) 4 (1978) 27–53.

    Google Scholar 

  16. D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data”, in:Proceedings of the 23rd National Conference ACM (ACM, New York, 1965) pp. 517–524.

    Google Scholar 

  17. A. Žilinskas, “Ob aksiomaticheskoj charakterizaciji statisticheskich modelej mnogoekstremalnych funkcij”, in:Primenenija sluchainogo poiska v prakticheskich zadachach (preprint VINITI, Moscow, 1980) pp. 106–110.

  18. A. Katkauskaite, “Sluchajnyje polja s nezavisimymi prirashchenijami”, Litovskii Matematičeskii 4 (1972) 75–85.

    Google Scholar 

  19. A. Žilinskas and E. Senkiene, “Ob ocenivaniji parametra vinerovskogo procesa”, Litovskii Matematičeskii 3 (1978) 59–62.

    Google Scholar 

  20. A. Žilinskas and E. Senkiene, “Ob ocenke parametra vinerovskogo sluchajnogo polja po rezultatam nabljudenija v sluchajnych zavisimych tochkach”,Kibernetika (Kiev) 6 (1979) 107–109. [Translated as: “On the estimation of parameter of the Wiener random field from the observations at random dependent points”,Cybernetics].

    Google Scholar 

  21. P. Fishburn,Utility theory for decision making (Wiley, New York, 1970).

    Google Scholar 

  22. A. Žilinskas, “The use of statistical models for construction of multimodal optimization algorithms”, in:Third Czechoslovak—Soviet—Hungarian Seminar on Information Theory (Czechoslovak Academy of Sciences, Prague, 1980) pp. 219–224.

    Google Scholar 

  23. A. Žilinskas, “Two algorithms for one-dimensional multimodal minimization”,Mathematische Operationsforschung und Statistik, Series Optimization 12 (1981) 53–63.

    Google Scholar 

  24. A. Torn, “A search approach to global optimization”, in: L.C.W. Dixon and G.P. Szego, eds.,Towards global optimization 2 (North-Holland, Amsterdam, 1978) pp. 49–62.

    Google Scholar 

  25. A. Žilinskas, “On the use of statistical models of multimodal functions for the construction of the optimization algorithms”, in: A. Balakrishnan and M. Thoma, eds.,Lecture notes in control and information sciences, Vol. 23 (Springer Verlag, Berlin, 1980) pp. 138–147.

    Google Scholar 

  26. W. Gordon and A. Wixom, “Shepard's method of ‘Metric Interpolation’”,Mathematics of Computation 39 (1978) 253–264.

    Google Scholar 

  27. A. Žilinskas, “On one-dimensional multimodal minimization”, in:Transactions of 8th Prague Conference on information theory, statistical decision functions and random processes, v. B (Academia, Prague, 1978) pp. 398–402.

    Google Scholar 

  28. A. Žilinskas, “Optimization of one-dimensional multimodal functions, Algorithm AS 133”,Applied Statistics 27 (1978) 367–375.

    Google Scholar 

  29. L.C. Dixon and G.P. Szego, “The global optimization problem: an introduction”, in L.C.W. Dixon and G.P. Szego, eds.,Towards global optimization 2 (North-Holland, Amsterdam, 1978) pp. 1–15.

    Google Scholar 

  30. A. Žilinskas, “Mimun-optimization of one-dimensional multimodal functions in the presence of noise, Algoritmus 44”,Aplikace Matematiky 25 (1980) 234–240.

    Google Scholar 

  31. H. Kushner, “A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise”,Transactions of the ASME series D 86 (1964) 97–105.

    Google Scholar 

  32. V. Šaltenis, “Ob odnom metode mnogoekstremalnoj optimizaciji”,Avtomatika i Vyčislitelnaja Tehnika (Riga) 3 (1971) 53–62.

    Google Scholar 

  33. A. Žilinskas, “Odnoshagovyj metod poiska ekstremuma funkciji odnoj peremenoj”,Kibernetika (Kiev) 1 (1975) 139–144.

    Google Scholar 

  34. R. Strongin,Chyslenyje metody v mnogoekstremalnych zadachach (Nauka, Moscow, 1978).

    Google Scholar 

  35. F. Archetti, “A probabilistic algorithm for global optimization problem with a dimensional reduction technique”, in: A. Balakrishnan and M. Thomas, eds.,Lecture notes in control and information sciences, Vol. 23 (Springer Verlag, Berlin, 1980) pp. 36–42.

    Google Scholar 

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  1. Institute of Mathematics and Cybernetics, Academy of Sciences, Vilnius, Lithuania, USSR

    A. Žilinskas

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Žilinskas, A. Axiomatic approach to statistical models and their use in multimodal optimization theory. Mathematical Programming 22, 104–116 (1982). https://doi.org/10.1007/BF01581029

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  • DOI: https://doi.org/10.1007/BF01581029

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