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Randomized algorithms in interval global optimization

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Abstract

This paper is a critical survey of the interval optimization methods aimed at computing global optima for multivariable functions. To overcome some drawbacks of traditional deterministic interval techniques, we outline some ways of constructing stochastic (randomized) algorithms in interval global optimization, in particular, those based on the ideas of random search and simulated annealing.

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References

  1. Azencott, R., Protsedura “otpuska” (Simulated “Annealing”), Proc. Sem. Bourbaki, 1988, Moscow: Mir, 1990, pp. 235–251.

    Google Scholar 

  2. Alefeld, G. and Herzberger, J., Vvedenie v intervalnye vychisleniya (Introduction to Interval Computations), Moscow: Mir, 1987.

    Google Scholar 

  3. Gaganov, A.A.,On Complexity of Calculating the Range of a Multivariable Polynomial, Kibern., 1985, no. 4, pp. 6–8.

  4. Evtushenko, Yu.G. and Rat’kin, V.A., Method of Bisection for Global Optimization of Multivariable Function, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1987, no. 1, pp. 119–128.

  5. Zhiglyavskii, A.A. and Zhilinskas, A.G., Metody poiska global’nogo ekstremuma (Methods of Search for a Global Extremum), Moscow: Nauka, 1991.

    Google Scholar 

  6. Intervalnyi analiz i ego prilozheniya (Interval Analysis and Its Applications); http://www.nsc.ru/interval/.

  7. Kalmykov, S.A., Shokin, Yu.I., and Yuldashev, Z.Kh., Metody interval’nogo analiza (Methods of Interval Analysis), Novosibirsk: Nauka, 1986.

    MATH  Google Scholar 

  8. Panov, N.V. and Koldakov, V.V., Programmnyi Kompleks dlya Graficheskogo Predstavleniya Protsessa i Rezul’tatov Raboty Interval’nykh Algoritmov, Trudy 5-oi mezhdunarodnoi konferentsii “Perspectivy sistem informatiki” pamyati akad. A.P. Ershova (A. Ershov Fifth International Conference on Perspectives for System Informatics), Novosibirsk, 2003, pp. 38–45.

  9. Panov, N.V. and Shary, S.P., Stochastic Approaches to Interval Methods of Global Optimization, Vserossiiskoe soveshchanie po interval’nomu analizu i ego prilozheniyam INTERVAL-06 (All-Russian Meeting on Interval Analysis and Its Applications INTERVAL-06), St.-Petersburg, 2006, pp. 101–105.

  10. Rastrigin, L.A., Statisticheskie metody poiska (Statistical Methods of Search),Moscow: Nauka, 1968.

    Google Scholar 

  11. Shary, S.P., Stochastic Approaches to Interval Global Optimization, Trudy 13-oi Baikal’skoi mezhdunar. shkoly-seminara “Metody optimizatsii i ikh prilozheniya” (Proc. 13th Baikal Intern. Workshop on Optimization Methods and Their Applications), Irkutsk, 2005, vol. 4., pp. 85–105.

    Google Scholar 

  12. Aarts, E. and Korst, J., Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing, Chichester: J. Wiley & Sons, 1989.

    MATH  Google Scholar 

  13. Corliss, G.F. and Kearfott, R.B., Rigorous Global Search: Industrial Applications, in Developments in Reliable Computing, Csendes, T., Ed., Dordrecht: Kluwer, 1999, pp. 1–16; http://interval.louisiana.edu/preprints/scan98.pdf/.

    Google Scholar 

  14. Dixon, L.C. and Szegö, G.P., The Global Optimization Problem: An Introduction, in Towards Global Optimization II, Dixon, L.C. and Szegö, G.P., Eds., Amsterdam: North Holland, 1978, pp. 1–15.

    Google Scholar 

  15. Encyclopedia of Optimization, Floudas, C.A. and Pardalos, P.M., Eds., Dordrecht: Kluwer, 2001, vols. I–VI.

    MATH  Google Scholar 

  16. Hansen, E. and Walster, G.W., Global Optimization Using Interval Analysis, New York: Marcel Dekker, 2004.

    MATH  Google Scholar 

  17. Kearfott, R.B., Rigorous Global Search: Continuous Problems, Dordrecht: Kluwer, 1996.

    MATH  Google Scholar 

  18. Kreinovich, V. and Kearfott, R.B., Beyond Convex? Global Optimization Is Feasible Only for Convex Objective Functions: A Theorem, J. Glob. Optim., 2005, vol. 33, no. 4, pp. 617–624.

    Article  MATH  MathSciNet  Google Scholar 

  19. Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., Optimization by Simulated Annealing, Science, 1983, vol. 220, pp. 671–680.

    Article  MathSciNet  Google Scholar 

  20. Moore, R.E., Methods and Applications of Interval Analysis, Philadelphia: SIAM, 1979.

    MATH  Google Scholar 

  21. Neumaier, A., Interval Methods for Systems of Equations, Cambridge: Cambridge Univ. Press, 1990.

    MATH  Google Scholar 

  22. Ratschek, H. and Rokne, J., New Computer Methods for Global Optimization, New York: Halsted Press, 1988.

    MATH  Google Scholar 

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Correspondence to S. P. Shary.

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Original Russian Text © S.P. Shary, 2008, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2008, Vol. 11, No. 4, pp. 457–474.

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Shary, S.P. Randomized algorithms in interval global optimization. Numer. Analys. Appl. 1, 376–389 (2008). https://doi.org/10.1134/S1995423908040083

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  • DOI: https://doi.org/10.1134/S1995423908040083

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