Minimizing noisy functionals in Hilbert space: An extension of the Kiefer-Wolfowitz procedure (Q1103307)

The Kiefer-Wolfowitz stochastic approximation procedure can be used for locating of an extremum of a real valued functional g defined on a Hilbert space H. An extension of this procedure for the case of infinite- dimensional Hilbert spaces is presented. The method of random directions is used in such a way that at iteration n attention is restricted to a subspace of finite-dimension \(k_ n\) where \(k_ n\to \infty\). Some necessary restrictions on the functional g must be supposed, the most serious being that in each finite dimensional subspace the functional g should have a ``pseudo-extremum'' and that the sequence of this ``pseudo-extrema'' tends to the ``global'' extremum of g in H. The main convergence theorem is presented and proved. Applications of this procedure to optimal stochastic control are given.