ERGODIC CONVERGENCE OF A STOCHASTIC PROXIMAL POINT ALGORITHM

The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates (x(n)) given by x(n+1) = (I+lambda(n) A(xi(n+1),.)) (1)(x(n)), where (A(s, .) : s is an element of E) is a collection of maximal monotone operators on a separable Hilbert space, (xi(n)) is an independent identically distributed sequence of random variables on E and (lambda(n)) is a positive sequence in l(2/)l(1). The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation E(A(xi(1) , .)) under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form E(f(xi(1,) x) w.r.t. x is an element of boolean AND(m)(n=i) X-i,X- where f is a normal convex integrand and (X-i) is a collection of closed convex sets. In this case, the iterations are closely related to a stochastic proximal algorithm recently proposed by Wang and Bertsekas.