Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem

We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the domi...

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Published in:	Journal of optimization theory and applications Vol. 178; no. 3; pp. 726 - 742
Main Authors:	Haskell, William B., Toriello, Alejandro
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2018 Springer Nature B.V
Subjects:	Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Constraints Dominance Economic models Engineering Mathematics Mathematics and Statistics Nonlinear programming Operations Research/Decision Theory Optimization Probability theory Theorems Theory of Computation Stochastic dominance Strassen theorem 90C15 Convex optimization 90C25
ISSN:	0022-3239, 1573-2878
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Abstract	We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.
AbstractList	We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.
Author	Haskell, William B. Toriello, Alejandro
Author_xml	– sequence: 1 givenname: William B. orcidid: 0000-0002-9518-4310 surname: Haskell fullname: Haskell, William B. email: isehwb@nus.edu.sg organization: National University of Singapore – sequence: 2 givenname: Alejandro surname: Toriello fullname: Toriello, Alejandro organization: Georgia Institute of Technology
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Cites_doi	10.1007/s10479-013-1337-0 10.1007/978-1-4612-1394-9 10.1007/s10107-006-0722-8 10.1137/08074009X 10.1137/1.9781611970524 10.1137/S1052623497315768 10.1016/j.orl.2005.10.004 10.1007/s10107-007-0165-x 10.1016/j.ejor.2011.11.019 10.1016/j.dam.2009.07.008 10.1007/s11067-011-9167-6 10.1007/s10107-009-0270-0 10.1016/j.jbankfin.2005.04.024 10.1137/070679569 10.1007/978-3-642-04900-2 10.1007/978-1-4612-0561-6 10.1016/j.orl.2008.05.003 10.1137/070707956 10.1007/s10107-009-0326-1 10.1007/s10107-010-0428-9 10.1007/978-0-387-34675-5 10.1137/120874679 10.1080/0740817X.2014.889336 10.1137/060650118 10.1214/aoms/1177700153 10.1515/9781400873173 10.1007/s10107-003-0453-z 10.1007/s10107-009-0321-6
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Copyright	Springer Science+Business Media, LLC, part of Springer Nature 2018 Journal of Optimization Theory and Applications is a copyright of Springer, (2018). All Rights Reserved.
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References	NieYWuXHomem-de MelloTOptimal path problems with second-order stochastic dominance constraintsNetw. Spat. Econ.2012124561587300465510.1007/s11067-011-9167-61332.90184 LizyayevARuszczyńskiATractable almost stochastic dominanceEur. J. Oper. Res.20122182448455287446410.1016/j.ejor.2011.11.0191244.91112 Armbruster, B., Delage, E.: Decision making under uncertainty when preference information is incomplete. Working Paper (2011) RockafellarRTConvex Analysis1970PrincetonPrinceton University Press10.1515/97814008731730193.18401 ArmbrusterBLuedtkeJModels and formulations for multivariate dominance constrained stochastic programsIIE Trans.201447111410.1080/0740817X.2014.889336 DentchevaDHenrionRRuszczyńskiAStability and sensitivity of optimization problems with first order stochastic dominance constraintsSIAM J. Optim.2007181322337229968710.1137/0606501181160.90607 DentchevaDRuszczyńskiARobust stochastic dominance and its application to risk-averse optimizationMath. Program.201012585100257732310.1007/s10107-009-0321-61216.90064 Armbruster, B.: A short proof of Strassen’s theorem using convex analysis. (2016). arXiv:1603.00137 DentchevaDRuszczyńskiAStochastic dynamic optimization with discounted stochastic dominance constraintsSIAM J. Control Optim.200847525402556244847310.1137/0706795691180.90211 DentchevaDRuszczyńskiAOptimality and duality theory for stochastic optimization problems with nonlinear dominance constraintsMath. Program.200499329350203904410.1007/s10107-003-0453-z1098.90044 GollmerRGotzesUSchultzRA note on second-order stochastic dominance constraints induced by mixed-integer linear recourseMath. Program.20111261179190276434410.1007/s10107-009-0270-01229.90109 HaskellWJainRStochastic dominance-constrained Markov decision processesSIAM J. Control Optim.2013511273303303287610.1137/1208746791330.90126 de MelloTHMehrotraSA cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraintsSIAM J. Optim.200920312501273254634310.1137/08074009X1198.90291 DentchevaDRuszczyńskiAPortfolio optimization with stochastic dominance constraintsJ. Bank. Finance200630243345110.1016/j.jbankfin.2005.04.0241206.90108 ShakedMShanthikumarJStochastic Orders2007BerlinSpringer10.1007/978-0-387-34675-50806.62009 Hernandez-LermaOLasserreJBApproximation schemes for infinite linear programsSIAM J. Optim.199884973988164128210.1137/S10526234973157680912.90219 DentchevaDRuszczyńskiAOptimization with stochastic dominance constraintsSoc. Ind. Appl. Math. J. Optim.200314254856620481551055.90055 LuedtkeJNew formulations for optimization under stochastic dominance constraintsSIAM J. Optim.20081914331450246617810.1137/0707079561180.90215 BotRIConjugate Duality in Convex Optimization2010BerlinSpringer10.1007/978-3-642-04900-21190.90002 Hernandez-LermaOLasserreJBFurther Topics On Discrete-Time Markov Control Processes1999New YorkSpringer10.1007/978-1-4612-0561-60928.93002 BonnansJShapiroAPerturbation Analysis of Optimization Problems2000New YorkSpringer10.1007/978-1-4612-1394-90966.49001 WangWAhmedSSample average approximation of expected value constrained stochastic programsOper. Res. Lett.2008365515519245813010.1016/j.orl.2008.05.0031210.90131 DrapkinDSchultzRAn algorithm for stochastic programs with first-order dominance constraints induced by linear recourseDiscrete Appl. Math.20101584291297258811210.1016/j.dam.2009.07.0081185.90160 Hu, J., Mehrotra, S.: Robust decision making using a general utility set. Technical Report, Working Paper, Department of IEMS, Northwestern University (2012) Hu, J., Mehrotra, S.: Robust decision making using a risk-averse utility set. Technical Report, Working Paper, Department of Industrial Engineering and Management Sciences, Northwestern University (2012) StrassenVThe existence of probability measures with given marginalsAnn. Math. Stat.196536242343917743010.1214/aoms/11777001530135.18701 AliprantisCBorderKInfinite Dimensional Analysis: A Hitchhiker’s Guide2006BerlinSpringer1156.46001 RockafellarRTConjugate Duality and Optimization1974PhiladelphiaSIAM10.1137/1.97816119705240296.90036 DurrettRProbability: Theory and Examples2005GroveThomson Brooks/Cole1202.60002 MüllerAStoyanDComparison Methods for Stochastic Models and Risks2002HobokenWiley0999.60002 RomanDDarby-DowmanKMitraGPortfolio construction based on stochastic dominance and target return distributionsMath. Program.20061082541569223871410.1007/s10107-006-0722-81138.91476 DentchevaDRuszczyńskiAOptimization with multivariate stochastic dominance constraintsMath. Program.2009117111127242130110.1007/s10107-007-0165-x1221.90069 NoyanNRudolfGRuszczyńskiARelaxations of linear programming problems with first order stochastic dominance constraintsOper. Res. Lett.2006346653659225151510.1016/j.orl.2005.10.0041112.90053 FábiánCMitraGRomanDProcessing second-order stochastic dominance models using cutting-plane representationsMath. Program.201113013357285315910.1007/s10107-009-0326-11229.90108 HuJHomem-de MelloTMehrotraSSample average approximation of stochastic dominance constrained programsMath. Program.20121331–2171201292109610.1007/s10107-010-0428-91259.90083 HaskellWShenZShanthikumarJOptimization with a class of multivariate integral stochastic order constraintsAnn. Oper. Res.20132061147162307384210.1007/s10479-013-1337-01301.90092 W Haskell (1339_CR18) 2013; 206 C Fábián (1339_CR11) 2011; 130 O Hernandez-Lerma (1339_CR35) 1999 B Armbruster (1339_CR25) 2014; 47 R Gollmer (1339_CR12) 2011; 126 D Dentcheva (1339_CR4) 2007; 18 A Müller (1339_CR1) 2002 C Aliprantis (1339_CR36) 2006 W Wang (1339_CR28) 2008; 36 A Lizyayev (1339_CR17) 2012; 218 M Shaked (1339_CR2) 2007 D Dentcheva (1339_CR6) 2003; 14 RT Rockafellar (1339_CR32) 1974 TH Mello de (1339_CR15) 2009; 20 D Dentcheva (1339_CR13) 2010; 125 D Roman (1339_CR9) 2006; 108 RT Rockafellar (1339_CR30) 1970 V Strassen (1339_CR24) 1965; 36 D Dentcheva (1339_CR19) 2008; 47 Y Nie (1339_CR10) 2012; 12 RI Bot (1339_CR31) 2010 R Durrett (1339_CR29) 2005 D Dentcheva (1339_CR14) 2009; 117 J Hu (1339_CR16) 2012; 133 N Noyan (1339_CR3) 2006; 34 W Haskell (1339_CR20) 2013; 51 1339_CR21 1339_CR23 1339_CR22 D Drapkin (1339_CR5) 2010; 158 D Dentcheva (1339_CR8) 2006; 30 D Dentcheva (1339_CR7) 2004; 99 J Luedtke (1339_CR26) 2008; 19 1339_CR27 O Hernandez-Lerma (1339_CR34) 1998; 8 J Bonnans (1339_CR33) 2000
References_xml	– reference: ArmbrusterBLuedtkeJModels and formulations for multivariate dominance constrained stochastic programsIIE Trans.201447111410.1080/0740817X.2014.889336 – reference: DentchevaDRuszczyńskiAOptimality and duality theory for stochastic optimization problems with nonlinear dominance constraintsMath. Program.200499329350203904410.1007/s10107-003-0453-z1098.90044 – reference: de MelloTHMehrotraSA cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraintsSIAM J. Optim.200920312501273254634310.1137/08074009X1198.90291 – reference: RockafellarRTConjugate Duality and Optimization1974PhiladelphiaSIAM10.1137/1.97816119705240296.90036 – reference: GollmerRGotzesUSchultzRA note on second-order stochastic dominance constraints induced by mixed-integer linear recourseMath. Program.20111261179190276434410.1007/s10107-009-0270-01229.90109 – reference: DentchevaDRuszczyńskiARobust stochastic dominance and its application to risk-averse optimizationMath. Program.201012585100257732310.1007/s10107-009-0321-61216.90064 – reference: Hernandez-LermaOLasserreJBApproximation schemes for infinite linear programsSIAM J. Optim.199884973988164128210.1137/S10526234973157680912.90219 – reference: BotRIConjugate Duality in Convex Optimization2010BerlinSpringer10.1007/978-3-642-04900-21190.90002 – reference: DentchevaDHenrionRRuszczyńskiAStability and sensitivity of optimization problems with first order stochastic dominance constraintsSIAM J. Optim.2007181322337229968710.1137/0606501181160.90607 – reference: LuedtkeJNew formulations for optimization under stochastic dominance constraintsSIAM J. Optim.20081914331450246617810.1137/0707079561180.90215 – reference: ShakedMShanthikumarJStochastic Orders2007BerlinSpringer10.1007/978-0-387-34675-50806.62009 – reference: Hu, J., Mehrotra, S.: Robust decision making using a risk-averse utility set. Technical Report, Working Paper, Department of Industrial Engineering and Management Sciences, Northwestern University (2012) – reference: LizyayevARuszczyńskiATractable almost stochastic dominanceEur. J. Oper. Res.20122182448455287446410.1016/j.ejor.2011.11.0191244.91112 – reference: HaskellWShenZShanthikumarJOptimization with a class of multivariate integral stochastic order constraintsAnn. Oper. Res.20132061147162307384210.1007/s10479-013-1337-01301.90092 – reference: DentchevaDRuszczyńskiAStochastic dynamic optimization with discounted stochastic dominance constraintsSIAM J. Control Optim.200847525402556244847310.1137/0706795691180.90211 – reference: Armbruster, B.: A short proof of Strassen’s theorem using convex analysis. (2016). arXiv:1603.00137 – reference: FábiánCMitraGRomanDProcessing second-order stochastic dominance models using cutting-plane representationsMath. Program.201113013357285315910.1007/s10107-009-0326-11229.90108 – reference: RockafellarRTConvex Analysis1970PrincetonPrinceton University Press10.1515/97814008731730193.18401 – reference: NoyanNRudolfGRuszczyńskiARelaxations of linear programming problems with first order stochastic dominance constraintsOper. Res. Lett.2006346653659225151510.1016/j.orl.2005.10.0041112.90053 – reference: NieYWuXHomem-de MelloTOptimal path problems with second-order stochastic dominance constraintsNetw. Spat. Econ.2012124561587300465510.1007/s11067-011-9167-61332.90184 – reference: Hernandez-LermaOLasserreJBFurther Topics On Discrete-Time Markov Control Processes1999New YorkSpringer10.1007/978-1-4612-0561-60928.93002 – reference: BonnansJShapiroAPerturbation Analysis of Optimization Problems2000New YorkSpringer10.1007/978-1-4612-1394-90966.49001 – reference: AliprantisCBorderKInfinite Dimensional Analysis: A Hitchhiker’s Guide2006BerlinSpringer1156.46001 – reference: DurrettRProbability: Theory and Examples2005GroveThomson Brooks/Cole1202.60002 – reference: MüllerAStoyanDComparison Methods for Stochastic Models and Risks2002HobokenWiley0999.60002 – reference: DentchevaDRuszczyńskiAOptimization with stochastic dominance constraintsSoc. Ind. Appl. Math. J. Optim.200314254856620481551055.90055 – reference: HaskellWJainRStochastic dominance-constrained Markov decision processesSIAM J. 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Snippet	We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained...
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SubjectTerms	Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Constraints Dominance Economic models Engineering Mathematics Mathematics and Statistics Nonlinear programming Operations Research/Decision Theory Optimization Probability theory Theorems Theory of Computation
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Title	Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem
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