SIMULATED ANNEALING TYPE ALGORITHMS FOR MULTIVARIATE OPTIMIZATION

We study the convergence of a class of discrete-time continuous-state simulated annealing type algorithms for multivariate optimization. The general algorithm that we consider is of the form X(k + 1) = X(k) - a(k)(DELTA-U(X(k)) + zeta-k) + b(k)W(k). Here U(.) is a smooth function on a compact subset of R(d), {zeta-k} is a sequence of R(d)-valued random variables, {W(k)} is a sequence of independent standard d-dimensional Gaussian random variables, and {a(k)}, {b(k)} are sequences of positive numbers which tend to zero. These algorithms arise by adding decreasing white Gaussian noise to gradient descent, random search, and stochastic approximation algorithms. We show under suitable conditions on U(.), {zeta-k}, {a-k}, and {b-k} that X(k) converges in probability to the set of global minima of U(.). A careful treatment of how X(k) is restricted to a compact set and its effect on convergence is given.