Optimization Under First Order Stochastic Dominance Constraints

We consider stochastic optimization problems involving stochastic dominance constraints of first order, also called stochastic ordering constraints. They are equivalent to a continuum of probabilistic constraints or chance constraints. We develop first order necessary and sufficient conditions of optimality for these models. We show that the Lagrange multipliers corresponding to stochastic dominance constraints are piecewise constant nondecreasing utility functions. These results extend our theory of stochastic dominance-constrained optimization to the first order case, in which the main challenge is the potential non- convexity of the problem. We also show that the convexification of stochastic ordering relation is equivalent to second order stochastic dominance under rather weak assumptions. This paper appeared as "Semi-Infinite Probabilistic Optimization: First Order Stochastic Dominance Constraints" in "Optimization" 53(2004) 583-- 601.