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Methods of solution of nonlinear extremal problems

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Conclusion

In the present paper only general methods of solving mathematical programming problems have been considered, and special methods intended for solving problems of a definite class have not been touched upon. In particular, the method of dynamic programming has not been considered.

The development of simple special methods is of paramount practical value. As a rule, a concrete extremal problem has essential peculiarities, because of which it is possible to conduct a more meaningful investigation than is possible in the general case, and thus obtain a simplified method of solution.

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References

  1. I. R. Blum, Multidimensional stochastic approximation method, AMS, 25, 1954.

  2. I. R. Blum, A note on stochastic approximation, Proc. Am. Math. Soc., 9, 1958.

  3. H. F. Bohnenblust, Theory of Games; Modern Mathematics for the Engineer, E. F. Beckenbach, ed. McGraw-Hill Book Co., N. Y., 1956.

    Google Scholar 

  4. N. P. Buslenko and G. A. Sokolov, “On a class of optimal distribution problems,” zhurn. Ekonomika i matematicheskie metody, Moscow, no. 1, 1965.

  5. S. Vajda, “Theory of games and linear programming,” collection: Linear Inequalities and Related Problems [Russian translation], IL, Moscow, 1959.

    Google Scholar 

  6. V. A. Volkonskii, “Optimal planning methods with a large number of dimensions,” zhurn. Ekonomika i matematicheskie metody, Moscow, No. 2, 1965.

  7. P. Wolfe, New-Methods in Nonlinear Programming [in Russian], izd-vo Nauka, Moscow, vol. 3, 1965.

    Google Scholar 

  8. S. Gass, Linear Programming [Russian translation], Fizmatgiz, Moscow, 1961.

    Google Scholar 

  9. I. M. Gel'fand and M. L. Tseitlin, “Principle of nonlocal search in automatic optimization systems,” DAN SSSR, vol. 137, no. 2, 1961.

  10. E. G. Gol'shtein, “Dual problems of convex programming,” zhurn. Ekonomika i matematicheskie metody, Moscow, no. 3, 1965.

  11. G. B. Dennis, Mathematical Programming and Electric Networks [Russian translation], IL, Moscow, 1961.

    Google Scholar 

  12. G. Zoutendijk, Methods of Feasible Directions, Elsevier, Amsterdam, 1960.

    Google Scholar 

  13. S. Karlin, Mathematical Methods and Theory in Games, Programming, and Economics, Addison-Wesley Publ. Co., Reading, Mass., 1959.

    Google Scholar 

  14. Henry J. Kelley, “Gradient methods,” collection: Optimization Techniques with Application to Aerospace Systems [Russian translation], izd-vo Nauka, Moscow, 1965.

    Google Scholar 

  15. G. P. Kintsi and V. Krelle, Nonlinear Programming [in Russian], izd-vo Sovetskoe radio, Moscow, 1965.

    Google Scholar 

  16. I. Kiefer and I. Wolfowitz, Stochastic estimation of the maximum of a regression function, The Annals of Mathematical Statistics, 23, 1952.

  17. B. N. Pshenchniy, “Duality principle in convex programming problems,” Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki, Moscow, no. 1, 1965.

  18. L. A. Rastrigin, Random Search [in Russian], izd-vo Zinatne, Riga, 1965.

    Google Scholar 

  19. H. Robbins and F. Monro, Astochastic approximation method, The Annals of Mathematical Statistics, 22, 1951.

  20. C. B. Tompkins, “Methods of Steep Descent” in: Modern Mathematics for the Engineer, E. F. Beckenbach, ed., McGraw-Hill Book Co., N. Y., 1956.

    Google Scholar 

  21. M. Frank and P. Wolfe, “An algorithm for quadratic programming” Naval Research Logistics Quarterly terly, vol. 3, 1956.

  22. N. Z. Shor, On the Structure of Numerical Algorithms for Solving Optimal Planning and Projection Problems [in Russian] Author's abstract of dissertation, Institute of Cybernetics AS UkrSSR, 1964.

  23. K. J. Arrow, L. Hurwicz, and H. Uzawa (eds.) Studies in Linear and Nonlinear Programming, Stanford Univ. Press, Stanford, Cal., 1958.

    Google Scholar 

  24. R. Courant, Variational Method for the Solution of Problems of Equilibrium Soc., 49, 1943.

  25. M. V. Rybashov, “Gradient method of solving convex programming problems,” Avtomatika i telemekhanika, Moscow, no. 11, 1965.

  26. T. Pietrzykowski, On an iteration method for maximizing a concave function on a convex set, Prace ZAM, ser. A, N 13, 1961.

  27. Hoang T'ui, “Concave programming under linear constraints,” DAN SSSR, 159, 1964.

  28. R. E. Gomory and W. J. Baumol, “Integer programming and estimates,” collection: Numerical Methods of Optimal Programming [Russian translation], Novosibirsk, 1962.

  29. S. I. Zukhovitskii and L. I. Avdeeva, Linear and Convex Programming [in Russian], izd-vo Nauka, 1964.

  30. E. G. Gol'shtein and D. B. Yudin, New Directions in Linear Programming [in Russian], izd-vo Sovetskoe radio, Moscow, 1966.

    Google Scholar 

  31. I. I. Eremin, “Relaxation method of solving a system of inequalities with convex functions in left parts,” DAN SSSR, vol. 160, no. 5, 1965.

  32. T. S. Motzkin and I. I. Schoenberg, The relaxation method for linear inequalities, Canad. Journ. Math., 6, no. 3, 1954.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Cybernetics, AS UkrSSR, USSR

    Yu. M. Ermol'ev

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  1. Yu. M. Ermol'ev
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Kibernetika, Vol. 2, No. 4, pp. 1–17, 1966

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Ermol'ev, Y.M. Methods of solution of nonlinear extremal problems. Cybern Syst Anal 2, 1–14 (1966). https://doi.org/10.1007/BF01071403

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