1~ 2. 3. 4. 5. LITERATURE CITED R. M. Karp, "Convergence of combinatorial problems," Cybernetic Collection [Russian translation], New Series, No. 12, Mir, Moscow (1975). G. M. Adel'son-Vel'skii, E. A. Dinits, and A. V. Karzanov, Flux Algorithms [in Russian], Nauka, Moscow (1975). G. O. Wesolowskyand R. F. Love, "A nonlinear approximation method for solving a generalized rect- angular distance Weber problem," Manage. Sei., 18, No. 11 (1972). H. Juel and R. F. Love, "An efficient computational procedure for solving the multifaeility rectilinear facilities location problem," Oper. Res. Q., 2__77,No. 3 (1976). L. R. Ford, Jr., and D. R. Fulkerson, Flows in Networks, Princeton (1962). STOCHASTIC PROGRAMMING IN HILBERT SPACE Yu. M. Kaniovskii, P. S. Knopov, and Z. V. Nekrylova UDC 519.21 On-line planning, filtration, and forecasting have given rise to a need for stochastic optimization and estimation methods for spaces of infinite dimensions. Here we examine one such technique: a method of pro- jection type for stochastic quasigradients. 1. Convergence with Probability 1 We assume that we have a sequence of independent random elements {~(s, x)}~ given upon a stochastic space {~, F, P} and that take values from areaI separable Hilbert space H. We assume that the ~(s, x) are Borelian vector functions for s -> 1 upon x ~ X ~ H, where X is a convex closed set, and for which the follow- ing relations apply: M~ (s, x) = cJ~ (x) + %w (s, x), (1) ~(s, x) -- bl[ (s, x) -- ~z (s, x), (2) co 0o where {Cs}l , {as} 1 , {r are positive numerical sequences infc s > 0, c-~'/2a,--~ 0 ; {fs(X)}~, {w(s, x)}? are $ --~ r sequences of H-valued vector functions given on X, and { z (s, x)}~ are independent centered H-valued random * eX and that there exists a finite point x* vectors. We assume that fs (x) becomes zero at the urAque point x s such that Ii x~--x*ll~< I~+ --*0 (it follows from the closure of X that x* ~ X). To determine x* we consider the sequence {x s }~ defined by the following recurrence relation: 2 ' x~+'=~x(x'--[~+u(s' )~)' (3) x~EX, Mllx~lr-< o~, where the step factor is g, > 0, s > 1+ ~++ 0, and the normalization factor u(s, x) is the Borelian function x ~ X $.~0o and inf inf u(s, x)> 0 for x ~ X, ~s = ~(s, xS), '~X(" ) is the operator for projection on X, with xl independent of s>~l Ilxll.</+ oo { +(s, x)} 1 . We denote by k some nonnegative constants that are not necessarily identical; the inequalities between the random quantities are to be understood as almost certain. THEOREM 1. Let the following conditions hold: 1) u(s, x)(ll fs(x) ll + Ilw(s, x)]l) -< k(1 + IIx II), uZ(s, x)M IIz(s, x) ll2 <- k(1 + tlx It 2 + a[ ItX-Xstl2), as >- c~ > 0, s--l; Translated from Kibernetika, No. 6, pp. 71-79, November-Decem--b-~, 1978. Original article submitted" May 19, 1977. 878 0011-4235/78/1406-0878507.50 9 1979 Plenum Publishing Corporation