Relaxations of linear programming problems with first order stochastic dominance constraints

Noyan, Nilay; Rudolf, Gábor; Ruszczyński, Andrzej

doi:10.1016/j.orl.2005.10.004

Cited by 37 publications

(25 citation statements)

References 12 publications

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“…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…”

Section: Review Of Existing Resultsmentioning

confidence: 99%

“…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].…”

Section: E[h(w )] ≥ E[h(y )]mentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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New Formulations for Optimization under Stochastic Dominance Constraints

Luedtke¹

2008

SIAM J. Optim.

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Stochastic dominance constraints allow a decision-maker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first and second-order stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, and relaxing integrality in the first-order formulation yields a second-order formulation, demonstrating the tightness of this formulation. We also present a specialized branching strategy and heuristics which can be used with the new first-order formulation. Computational tests illustrate the potential benefits of the new formulations.

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“…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…”

Section: Review Of Existing Resultsmentioning

confidence: 99%

Section: E[h(w )] ≥ E[h(y )]mentioning

confidence: 99%

See 1 more Smart Citation

New Formulations for Optimization under Stochastic Dominance Constraints

Luedtke¹

2008

SIAM J. Optim.

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“…Because this formulation includes all the variables and constraints of the SSDLP formulation, the linear programming relaxation yields a bound at least as strong as the bound obtained from the SSD relaxation, relaxing the FSD constraint W 1 Y to an SSD constraint W 2 Y . For scalar stochastic dominance constraints it has been shown that the SSD relaxation is often good (Noyan et al, 2006). Furthermore, for an important special case in which W and Y take on all values with equal probability, we show in § 4.4 that the SSD relaxation is in a sense as good as can be achieved.…”

Section: Two Compact Mip Formulation (Fsd1 and Fsd1+ssd)mentioning

confidence: 62%

Models and formulations for multivariate dominance-constrained stochastic programs

Armbruster

Luedtke

2014

IIE Transactions

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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.

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“…This fact leads to an LP relaxation which is different from the usual one obtained by relaxing the integrality restriction. In [28] we provide examples which show that neither of these relaxations is stronger than the other.…”

Section: Second Order Constraints As Valid Inequalitiesmentioning

confidence: 92%

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

Noyan

Ruszczyński

2007

Math. Program.

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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.

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Relaxations of linear programming problems with first order stochastic dominance constraints

Cited by 37 publications

References 12 publications

New Formulations for Optimization under Stochastic Dominance Constraints

New Formulations for Optimization under Stochastic Dominance Constraints

Models and formulations for multivariate dominance-constrained stochastic programs

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

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