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

Abstract

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.

New Formulations for Optimization under Stochastic Dominance Constraints

“…Condition (7) can be used to derive a MIP formulation for a first-order stochastic dominance (FSD) constraint [24,25]. W (1) Y if and only if there exists β such that…”

“…For example, if other constraints in the model imply w i ≥ l i , then we can take M ik = y k − l i . Although this formulation was presented in [24,25], the authors do not recommend using this formulation for computation, since the linear programming relaxation bounds are too weak. Instead, because firstorder stochastic dominance implies second-order dominance, any formulation for second-order dominance is a relaxation of first order dominance, and the authors therefore propose to use the problem SSD as a relaxation for FSD.…”

“…In addition, problem SSD is used as a basis for heuristics to find feasible solutions for FSD. It is demonstrated in [25] that the bounds from using SSD as a relaxation of FSD are usually good, and that the heuristics are able to obtain good feasible solutions. However, these results do not yield a convergent algorithm for finding an optimal solution to FSD.…”

“…In addition, the linear relaxation of this formulation is also a formulation of SSD. As a result, the linear programming relaxation of this formulation is equivalent to the SSD relaxation proposed in [24], and shown to be a tight relaxation of FSD in [25].…”

“…In addition, dose-volume restrictions appearing in radiation treatment planning problems [18] can be formulated as a firstorder stochatic dominance constraint. Stochastic programming under stochastic dominance constraints has recently been studied in [4,5,6,8,11,12,24,25,26].…”