Empirical Estimates in Stochastic Programming

Abstract

One of the main problems in stochastic programming is to find estimates of the optimal value and the optimal solution. Solving practical problems we often need to solve some stochastic optimization problem without knowledge of the probability laws. From the mathematical point of view we have to associate with the original optimization problem an additional stochastic decision problem.

If the mathematical solution is sought with respect to the mathematical expectation then some approximate problems can be get setting an empirical distribution function instead of the theoretical one. The mentioned access was considered in the literature many times, Dupačová (1976), Kaňková (1974, 1978), and Цыбаков (1981) for example. There have been given rather general conditions under which the corresponding estimates converge to the theoretical values with probability one. Further, they studied the convergence rate. However, these results were based on the random samples with stochastic independent members only. The only exception was in Kaňková (1974).

In this paper we shall consider these approximate methods too. Especially, it will be the aim to study the estimates behaviour based on the random samples with stochastic dependent members.