"Classical" optimization problems depending on a probability measure belong mostly to nonlinear deterministic optimization problems that are, from the numerical point of view, relatively complicated. On the other hand, these problems fulfil very often assumptions giving a possibility to replace the "underlying" probability measure by an empirical one to obtain "good" empirical estimates of the optimal value and the optimal solution. Convergence rate of these estimates have been studied mostly for "underlying" probability measures with suitable (thin) tails. However, it is known that probability distributions with heavy tails better correspond to many economic problems. The paper focuses on distributions with finite first moments and heavy tails. The introduced assertions are based on the stability results corresponding to the Wasserstein metric with an "underlying" L1 norm and empirical quantiles convergence.
Optimization problems with stochastic dominance constraints are helpful to many real-life applications. We can recall e. g., problems of portfolio selection or problems connected with energy production. The above mentioned constraints are very suitable because they guarantee a solution fulfilling partial order between utility functions in a given subsystem U of the utility functions. Especially, considering U:=U1 (where U1 is a system of non decreasing concave nonnegative utility functions) we obtain second order stochastic dominance constraints. Unfortunately it is also well known that these problems are rather complicated from the theoretical and the numerical point of view. Moreover, these problems goes to semi-infinite optimization problems for which Slater's condition is not necessary fulfilled. Consequently it is suitable to modify the constraints. A question arises how to do it. The aim of the paper is to suggest one of the possibilities how to modify the original problem with an "estimation" of a gap between the original and a modified problem. To this end the stability results obtained on the base of the Wasserstein metric corresponding to L1 norm are employed. Moreover, we mention a scenario generation and an investigation of empirical estimates. At the end attention will be paid to heavy tailed distributions.
Optimization problems depending on a probability measure correspond to many applications. These problems can be static (single-stage), dynamic with finite (multi-stage) or infinite horizon, single- or multi-objective. It is necessary to have complete knowledge of the "underlying" probability measure if we are to solve the above-mentioned problems with precision. However this assumption is very rarely fulfilled (in applications) and consequently, problems have to be solved mostly on the basis of data. Stochastic estimates of an optimal value and an optimal solution can only be obtained using this approach. Properties of these estimates have been investigated many times. In this paper we intend to study one-stage problems under unusual (corresponding to reality, however) assumptions. In particular, we try to compare the achieved results under the assumptions of thin and heavy tails in the case of problems with linear and nonlinear dependence on the probability measure, problems with probability and risk measure constraints, and the case of stochastic dominance constraints. Heavy-tailed distributions quite often appear in financial problems \cite {Meer 2003)} while nonlinear dependence frequently appears in problems with risk measures \cite {Kan (2012a),Pflu (2007)}. The results we introduce follow mostly from the stability results based on the Wasserstein metric with the "underlying" L1 norm. Theoretical results are completed by a simulation investigation.