Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

Stochastic dominance relations are well-studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance (SSD) constraints can be solved by linear programming (LP). However, problems involving first order stochastic dominance (FSD) constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete FSD-constrained optimization model and present an LP relaxation based on SSD constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional SSD-constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a portfolio optimization problem.