Kybernetika

The assumption of a complete knowledge of the distribution in stochastic optimization problem is only seldom justified in real-life situations. Consequently, statistical estimates of the unknown probability measure, if they exist, can be only utilized to obtain some estimates of the optimal value and the optimal solution.

The empirical distribution function is usually used everywhere when the theoretical distribution function is fully unknown [1], [5], [17]. This substitution leads to the "good" statistical estimates [2], [9], [10], [14], [16]. However, unfortunately, it is also well-known that the corresponding approximative problem need not be a concave problem even in the case when the theoretical original one possesses this property. In particular, this happens rather often in the case of the chance constrained stochastic programming problems.

If we can assume that the theoretical distribution function belongs to a parametric family, then we can employ estimates of the unknown parameter to get some estimates of the optimal value and the optimal solution [3], [16]. In this paper, we shall consider the case when the unknown parameter can be introduced as a location parameter. We obtain the estimates of the optimal value and the optimal solution with statistical properties fully determined by the properties of the original parameter estimates. Moreover, the approximative problems belong to the same type of the optimization problems as the original one. However, to obtain these results we have to study the stability problem with respect to the location parameter, first.

At the end of the paper we shall try to apply some obtained results to stochastic optimization problem considered with respect to the discrete time interval $1 \div N.$ Namely surely, the main importance of the former results will be found just in such dynamic models.