Semi-infinite probabilistic optimization: First-order stochastic dominance constrain

We consider stochastic optimization problems involving a continuum of probabilistic constraints. They are equivalent to stochastic dominance constraints of first order, frequently called stochastic ordering constraints. We develop necessary and sufficient conditions of optimality for these models. We show that the Lagrange multipliers corresponding to dominance constraints can be identified with piecewise constant nondecreasing utility functions. We also show that the convexification of stochastic ordering relation is equivalent to second-order stochastic dominance under rather weak assumptions.