Models and formulations for multivariate dominance-constrained stochastic programs

Abstract

Dentcheva and Ruszczyński recently proposed using a stochastic dominance constraint to specify risk preferences in a stochastic program. Such a constraint requires the random outcome resulting from one's decision to stochastically dominate a given random comparator. These ideas have been extended to problems with multiple random outcomes, using the notion of positive linear stochastic dominance. We propose a constraint using a different version of multivariate stochastic dominance. This version is natural due to its connection to expected utility maximization theory and is relatively tractable. In particular, we show that such a constraint can be formulated with linear constraints for the second-order dominance relation, and with mixedinteger constraints for the first-order relation. This is in contrast to a constraint on second-order positive linear dominance, for which no efficient algorithms are known. We tested these formulations in the context of two applications: budget allocation in a setting with multiple objectives and finding radiation treatment plans in the presence of organ motion.

Stochastic Programming with Multivariate Second Order Stochastic Dominance Constraints with Applications in Portfolio Optimization

“…In particular, even for the case of second-order stochastic dominance, which induces a convex feasible region, their algorithm requires global optimization of a nonconvex problem as a subproblem. Furthermore, Armbruster and Luedtke [1] proposed to use a different notion of multivariate stochastic dominance as a constraint in a stochastic optimization model. They derived an LP formulation for an SSD constraint which could be solved using off-the-shelf linear programming solvers.…”

“…Therefore we may regard our algorithm as an extension of theirs. Furthermore, the proposed numerical methods provides an alternative approach to the existing cutting surface method for multivariate stochastic dominance introduced by Homem-de-Mello and Mehrota [10] and the linearized method proposed by Armbruster and Luedtke [1].…”

“…In this paper we effectively extend the methods proposed in [25] to stochastic programs with multivariate second order stochastic dominance constraints. We propose an exact penalization scheme for such problems and solve the penalized problem by the level function method proposed by Lemarechal et al [17] and extended by Xu [27] and a modified cutting plane method and compare them to the cutting surface method proposed by Homem-de-Mello et al [10] and the linearized method proposed by Armbruster and Luedtke [1].…”

“…They showed that such an approach is much less restrictive than the deterministic weighted approach. More recently, Armbruster and Luedtke [1] derived a linear formulation for multivariate second order stochastic dominance which can be solved with off the shelf linear programming solvers.…”

“…The budget allocation model is inspired by the homeland security application of Hu et al [12] and the budget allocation example of Armbruster and Luedtke [1], in which a limited budget must be allocated to a set of possible projects, and the allocation must stochastically dominate a given benchmark. The proposed method proved to be more efficient in sense of CPU time when solving larger problems compared to the linearized method proposed in [1]. For the portfolio optimization problem, we use real world test data of three indices to set up backtest and out-of-sample test to inspect the performance of the generated portfolio and compare it to the benchmark portfolio and a portfolio strategy generated by Markowitz model.…”