Conclusion
The results described in this paper can be extended to the case of Hilbert spaces, with small changes due mostly to the replacement of the word “vector” by the word “element.” Of greater interest from the practical point of view is the generalization of the method (9)–(10) to extremal problems with conditions of the form
for which projection for some reason cannot be made (for example, if the functions Fi(x) are not defined analytically and only statistical estimates of their values can be obtained). A stochastic variant of the Arrow-Hurwitz method was proposed in [7] for the solution of such problems.
It is of particular importance to find new means, different from (13)–(14), of controlling the step length ϱ s. Apart from the fact that the control (13)–(14) has a rigid, programmed nature, it is determinate, i.e., it does not depend on the actual trajectory of the “descent.” The process (9)–(10) is random and defines a whole family of trajectories from the initial point to the minimum point. The control (13)–(14) is calculated once for the whole family, so sufficiently rapid convergence cannot be expected from it. For a stochastic method of the form (9)–(10), stochastic means of control (see [17]) depending on the previous history of the “descent” are more natural.
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Literature Cited
Yu. M. Ermol'ev and A. D. Tuniev, “Random Féjer and quasi-Féjer sequences” collection: The Theory of Optimal solutions [in Russian], izd. IK AN UkrSSR, no. 2, Kiev, 1968.
N. Z. Shor, Author's abstract of candidate's dissertation: On the Structure of Algorithms for the Numerical Solution of Problems in Optimal Planning and Design, IK AN UkrSSR, Kiev, 1964.
Yu. M. Ermol'ev, “Methods of solution of nonlinear extremal problems,” Kibernetika, no. 4, Kiev, 1966.
Yu. M. Ermol'ev and N. Z. Shor, “On the minimization of nondifferentiable functions,” Kibernetika, no. 2, Kiev, 1967.
N. Z. Shor, “The convergence of the generalized gradient method,” Kibernetika, no. 3, Kiev, 1968.
B. T. Polyak, “A general method of solving extremal problems,” DAN SSSR,174, no. 1, Moscow, 1967.
Yu. M. Ermol'ev and Z. V. Nekrylova, “The method of stochastic gradients and its applications,” collection: The Theory of Optimal Solutions [in Russian], izd. IK AN UkrSSR, no. 1, Kiev, 1967.
Yu. M. Ermol'ev and N. Z. Shor, “The method of random search for the two-stage problem in stochastic programming and its generalization,” Kibernetika, no. 1, Kiev, 1968.
Yu. M. Ermol'ev and A. D. Tuniev, “Certain direct methods in stochastic programming,” Kibernetika, no. 4, Kiev, 1968.
I. R. Blum, “Multidimensional stochastic approximation methods,” Annals of Mathematical Statistics,25, no. 4, 1954.
G. B. Dantzig and Madansky, “On the solution of two-stage linear programming problems under uncertainty,” Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, no. 1, 1961.
S. Agmon, “The relaxation method for linear inequalities,” Canad. J. Math.,6, no. 3, 1954.
I. I. Eremin, “Some iteration methods in convex programming,” Ekonomika i matem. metody, no. 6, Nauka, Moscow, 1966.
J. Doob, Stochastic Processes [Russian translation], IL, Moscow, 1956.
Yu. M. Ermol'ev and V. P. Gulenko, “A finite-difference method for problems in optimal controls,” Kibernetika, no. 3, Kiev, 1967.
Ya. Z. Tsypkin, Adaptation and Learning in Automatic Systems, Izd. Nauka, Moscow, 1968.
Yu. M. Ermol'ev, “On the convergence of random quasi-Féjer sequences” collection: The Theory of Optimal Solutions [in Russian], izd. IK AN USSR, Kiev, 1969.
Yu. M. Ermol'ev, “Random optimization and stochastic programming,” Trudy zimnei shkoly po matem. programmirovaniyu, Drogobych, January–February, 1968.
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Translated from Kibernetika, Vol. 5, No. 2, pp. 73–84, March–April, 1969.
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Ermol'ev, Y.M. On the method of generalized stochastic gradients and quasi-Féjer sequences. Cybern Syst Anal 5, 208–220 (1969). https://doi.org/10.1007/BF01071091
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DOI: https://doi.org/10.1007/BF01071091