RECURSIVE STOCHASTIC ALGORITHMS FOR GLOBAL OPTIMIZATION IN RD

An algorithm of the form X(k+1) = X(k) - a(k)(nabla U(X(k) + xi-k) + b(k) W(k), where U(.) is a smooth function on R(d), {xi-k} is a sequence of R(d)-valued random variables, {W(k)} is a sequence of independent standard d-dimensional Gaussian random variables, a(k) = A/k and b(k) = square-root B/ square-root k log log k for k large, is considered. An algorithm of this type arises by adding slowly decreasing white Gaussian noise to a stochastic gradient algorithm. It is shown, under suitable conditions on U(.), {xi-k}, A, and B, that X(k) converges in probability to the set of global minima of U(.). No prior information is assumed as to what bounded region contains a global minimum. The analysis is based on the asymptotic behavior of the related diffusion process dY(t) = -nabla U(Y(t))dt + c(t)dW(t), where W(.) is a standard d-dimensional Wiener process and c(t) = square-root C/square-root log t for t large.