Decision problems involving random outcomes require the application of a decision principle to make the corresponding optimization problems well defined. One of the established principles is the relation of stochastic dominance. It establishes a partial order among random outcomes, but it is very difficult to use practice.

The practice frequently resorts to mean--risk models, which use two measures of quality: the expected outcome and some measure of the uncertainty of the outcome, called the risk.

In this talk we shall discuss relations between stochastic dominance and mean--risk models. We shall show that central semi-deviations used as risk measures are in harmony with the stochastic dominance orders of the corresponding degrees.

Next, by exploiting duality relations of convex analysis we show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation of the second degree.

We provide stochastic linear programming formulations of these models and we illustrate our results on a portfolio problem involving 719 securities and return data from the last 12 years.