A computational method for stochastic optimization

In this paper a computational method to obtain the solution to an unconstrained stochastic optimization problem is presented. A conjugate gradient scheme is applied to generate a sequence of random variables approximating the optimum solution to the problem. Under certain conditions, this, sequence converges to the optimum solution in the mean-square sense. Since a random variable is completely defined by its probability distribution the stochastic optimization problem is treated as a problem of obtaining the probability distribution for the optimum solution. This can be achieved by generating probability distributions for the convergent sequence of random variables obtained in the conjugate gradient scheme. A computational scheme to obtain the probability-density function for the limiting random variable is derived by using finite sums of Gaussian density to approximate the probability densities for the convergent sequence of random variables.