A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse

We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-st...

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Published in:	Mathematical programming Vol. 126; no. 1; pp. 179 - 190
Main Authors:	Gollmer, Ralf, Gotzes, Uwe, Schultz, Rüdiger
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer-Verlag 01.01.2011 Springer Springer Nature B.V
Subjects:	Algorithms Applied sciences Blocking Calculus of Variations and Optimal Control; Optimization Combinatorics Distribution theory Dominance Equivalence Exact sciences and technology Full Length Paper Integer programming Integers Linear programming Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Perturbation methods Probability and statistics Probability theory and stochastic processes Random variables Sciences and techniques of general use Statistics Stochastic models Stochasticity Studies Theoretical 60E15 Stochastic dominance 90C15 Stochastic integer programming 90C11 Mixed-integer optimization Structured programming Partial ordering Mixed integer programming Probability distribution Set constraint Modeling Optimization Stochastic programming Integer programming Ordering Large scale Test power Probability measure
ISSN:	0025-5610, 1436-4646
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Abstract	We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-structured, mixed- integer linear programming equivalents to the dominance constrained stochastic programs are identified. For these models, a decomposition algorithm is proposed and tested with instances from power optimization.
AbstractList	We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-structured, mixed- integer linear programming equivalents to the dominance constrained stochastic programs are identified. For these models, a decomposition algorithm is proposed and tested with instances from power optimization. We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-structured, mixed- integer linear programming equivalents to the dominance constrained stochastic programs are identified. For these models, a decomposition algorithm is proposed and tested with instances from power optimization.[PUBLICATION ABSTRACT]
Author	Gollmer, Ralf Gotzes, Uwe Schultz, Rüdiger
Author_xml	– sequence: 1 givenname: Ralf surname: Gollmer fullname: Gollmer, Ralf organization: Department of Mathematics, University of Duisburg-Essen – sequence: 2 givenname: Uwe surname: Gotzes fullname: Gotzes, Uwe organization: Department of Mathematics, University of Duisburg-Essen – sequence: 3 givenname: Rüdiger surname: Schultz fullname: Schultz, Rüdiger email: schultz@math.uni-duisburg.de organization: Department of Mathematics, University of Duisburg-Essen
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Cites_doi	10.1007/978-3-0348-6328-5 10.1137/060650118 10.1016/j.orl.2005.10.004 10.1016/0012-365X(77)90028-0 10.1016/S0167-6377(98)00050-9 10.1137/060678051 10.1007/978-94-017-3087-7 10.1007/s10107-003-0453-z 10.1016/j.ijepes.2006.03.003 10.1137/S1052623402420528
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References_xml	– reference: HandschinE.NeiseF.NeumannH.SchultzR.Optimal operation of dispersed generation under uncertainty using mathematical programmingInt. J. Electr. Power Energy Syst.20062861862610.1016/j.ijepes.2006.03.003 – reference: RuszczyńskiA.Decomposition methods in stochastic programmingMath. Program.199779333353 – reference: CPLEX Callable Library 9.1.3, ILOG (2005) – reference: BillingsleyP.Convergence of Probability Measures1968New YorkWiley0172.21201 – reference: PrékopaA.Stochastic Programming1995DordrechtKluwer – reference: NoyanN.RudolfG.RuszczyńskiA.Relaxations of linear programming problems with first order stochastic dominance constraintsOper. Res. Lett.2006346536591112.9005310.1016/j.orl.2005.10.0042251515 – reference: DentchevaD.HenrionR.RuszczyńskiA.Stability and sensitivity of optimization problems with first order stochastic dominance constraintsSIAM J. Optim.2007183223371160.9060710.1137/0606501182299687 – reference: RuszczyńskiA.ShapiroA.Handbooks in Operations Research and Management Science, 10: Stochastic Programming2003AmsterdamElsevier – reference: CarøeC.C.SchultzR.Dual decomposition in stochastic integer programmingOper. Res. Lett.19992437451063.9003710.1016/S0167-6377(98)00050-91684909 – reference: BlairC.E.JeroslowR.G.The value function of a mixed integer program: IDiscrete Math.1977191211380364.9007410.1016/0012-365X(77)90028-0475841 – reference: BirgeJ.R.LouveauxF.Introduction to Stochastic Programming1997New YorkSpringer0892.90142 – reference: DentchevaD.RuszczyńskiA.Optimization with stochastic dominance constraintsSIAM J. Optim.2003145485661055.9005510.1137/S10526234024205282048155 – reference: BankB.GuddatJ.KlatteD.KummerB.TammerK.Non-linear Parametric Optimization1982BerlinAkademie – reference: Gollmer, R., Gotzes, U., Schultz, R.: Second-order stochastic dominance constraints induced by mixed-integer linear recourse, Preprint 644/2007, Department of Mathematics, University of Duisburg-Essen (2007); also available online from http://www.uni-duisburg.de/FB11/disma/preprints.shtml – reference: MüllerA.StoyanD.Comparison Methods for Stochastic Models and Risks2002ChichesterWiley0999.60002 – reference: DentchevaD.RuszczyńskiA.Optimality and duality theory for stochastic optimization with nonlinear dominance constraintsMath. Program.2004993293501098.9004410.1007/s10107-003-0453-z2039044 – reference: BillingsleyP.Probability and Measure1986New YorkWiley0649.60001 – reference: KallP.WallaceS.W.Stochastic Programming1994ChichesterWiley0812.90122 – reference: GollmerR.NeiseF.SchultzR.Stochastic programs with first-order dominance constraints induced by mixed-integer linear recourseSIAM J. Optim.2008195525711173.9049010.1137/0606780512425029 – reference: BankB.MandelR.Parametric Integer Optimization1988BerlinAkademie0643.90043 – volume-title: Non-linear Parametric Optimization year: 1982 ident: 270_CR1 doi: 10.1007/978-3-0348-6328-5 – volume-title: Parametric Integer Optimization year: 1988 ident: 270_CR2 – volume-title: Introduction to Stochastic Programming year: 1997 ident: 270_CR5 – volume: 18 start-page: 322 year: 2007 ident: 270_CR9 publication-title: SIAM J. Optim. doi: 10.1137/060650118 – volume: 34 start-page: 653 year: 2006 ident: 270_CR17 publication-title: Oper. Res. Lett. doi: 10.1016/j.orl.2005.10.004 – volume: 79 start-page: 333 year: 1997 ident: 270_CR19 publication-title: Math. Program. – volume: 19 start-page: 121 year: 1977 ident: 270_CR6 publication-title: Discrete Math. doi: 10.1016/0012-365X(77)90028-0 – volume: 24 start-page: 37 year: 1999 ident: 270_CR7 publication-title: Oper. Res. Lett. doi: 10.1016/S0167-6377(98)00050-9 – volume-title: Probability and Measure year: 1986 ident: 270_CR4 – ident: 270_CR8 – volume: 19 start-page: 552 year: 2008 ident: 270_CR13 publication-title: SIAM J. Optim. doi: 10.1137/060678051 – volume-title: Stochastic Programming year: 1995 ident: 270_CR18 doi: 10.1007/978-94-017-3087-7 – ident: 270_CR12 – volume: 99 start-page: 329 year: 2004 ident: 270_CR11 publication-title: Math. Program. doi: 10.1007/s10107-003-0453-z – volume: 28 start-page: 618 year: 2006 ident: 270_CR14 publication-title: Int. J. Electr. Power Energy Syst. doi: 10.1016/j.ijepes.2006.03.003 – volume-title: Stochastic Programming year: 1994 ident: 270_CR15 – volume: 14 start-page: 548 year: 2003 ident: 270_CR10 publication-title: SIAM J. Optim. doi: 10.1137/S1052623402420528 – volume-title: Handbooks in Operations Research and Management Science, 10: Stochastic Programming year: 2003 ident: 270_CR20 – volume-title: Comparison Methods for Stochastic Models and Risks year: 2002 ident: 270_CR16 – volume-title: Convergence of Probability Measures year: 1968 ident: 270_CR3
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Snippet	We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set...
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Title	A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse
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