Statistical robustness in utility preference robust optimization models

Utility preference robust optimization (PRO) concerns decision making problems where information on decision maker’s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited. A key assumption in...

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Vydáno v:	Mathematical programming Ročník 190; číslo 1-2; s. 679 - 720
Hlavní autoři:	Guo, Shaoyan, Xu, Huifu
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021 Springer Nature B.V
Témata:	Calculus of Variations and Optimal Control; Optimization Combinatorics Continuity (mathematics) Decision making Estimators Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Optimization models Perturbation Risk management Statistical analysis Theoretical 90C31 Qualitative statistical robustness 90C47 Quantitative statistical robustness 90C15 Uniform consistency PRO
ISSN:	0025-5610, 1436-4646
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Abstract	Utility preference robust optimization (PRO) concerns decision making problems where information on decision maker’s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited. A key assumption in the PRO models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution. In data-driven optimization, this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the PRO models based on perceived data are reliable. In this paper, we investigate the issue which is also known as qualitative robustness in the literature of statistics (Huber in Robust statistics, 3rd edn, Wiley, New York, 1981) and risk management (Krätschmer et al. in Finance Stoch 18:271–295, 2014). By utilizing the framework proposed by Krätschmer et al. (2014), we derive moderate sufficient conditions under which the optimal value and optimal solution of the PRO models are robust against perturbation of the exogenous uncertainty data, and examine how the tail behaviour of utility functions affects the robustness. Moreover, under some additional conditions on the Lipschitz continuity of the underlying functions with respect to random data, we establish quantitative robustness of the statistical estimators under the Kantorovich metric. Finally, we investigate uniform consistency of the optimal value and optimal solution of the PRO models. The results cover utility selection problems and stochastic optimization problems as special cases.
AbstractList	Utility preference robust optimization (PRO) concerns decision making problems where information on decision maker’s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited. A key assumption in the PRO models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution. In data-driven optimization, this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the PRO models based on perceived data are reliable. In this paper, we investigate the issue which is also known as qualitative robustness in the literature of statistics (Huber in Robust statistics, 3rd edn, Wiley, New York, 1981) and risk management (Krätschmer et al. in Finance Stoch 18:271–295, 2014). By utilizing the framework proposed by Krätschmer et al. (2014), we derive moderate sufficient conditions under which the optimal value and optimal solution of the PRO models are robust against perturbation of the exogenous uncertainty data, and examine how the tail behaviour of utility functions affects the robustness. Moreover, under some additional conditions on the Lipschitz continuity of the underlying functions with respect to random data, we establish quantitative robustness of the statistical estimators under the Kantorovich metric. Finally, we investigate uniform consistency of the optimal value and optimal solution of the PRO models. The results cover utility selection problems and stochastic optimization problems as special cases.
Author	Xu, Huifu Guo, Shaoyan
Author_xml	– sequence: 1 givenname: Shaoyan surname: Guo fullname: Guo, Shaoyan organization: School of Mathematical Sciences, Dalian University of Technology – sequence: 2 givenname: Huifu orcidid: 0000-0001-8307-2920 surname: Xu fullname: Xu, Huifu email: hfxu@se.cuhk.edu.hk organization: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong
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References_xml	– reference: KrätschmerVSchiedAZähleHComparative and statistical robustness for law-invariant risk measuresFinance Stoch.201418271295317740710.1007/s00780-013-0225-4 – reference: ZähleHA definition of statistical robustness for general point estimators, and examplesJ. Multivar. Anal.2016143123110.1016/j.jmva.2015.08.004 – reference: HuberPJRobust Statistics19813New YorkWiley10.1002/0471725250 – reference: KrätschmerVSchiedAZähleHQualitative and infinitesimal robustness of tail-dependent statistical functionalsJ. Multivar. Anal.20121033547282370610.1016/j.jmva.2011.06.005 – reference: FöllmerHSchiedAStochastic Finance: An Introduction in Discrete Time20113Berlinde Gruyter10.1515/9783110218053 – reference: HaskellWFuLDessoukMAmbiguity in risk preferences in robust stochastic optimizationEur. J. Oper. Res.2016254214225349447010.1016/j.ejor.2016.03.016 – reference: DelageEKuhnDWiesemannW“Dice”-sion making under uncertainty: when can a random decision reduce risk?Manag. Sci.2019653282330110.1287/mnsc.2018.3108 – reference: Embrechts, P., Schied, A., Wang, R.: Robustness in the optimization of risk measures (2018). arXiv:1809.09268 – reference: ShapiroADentchevaDRuszczyńskiALectures on Stochastic Programming: Modeling and Theory20142PhiladelphiaSIAM10.1137/1.9781611973433 – reference: Claus, M.: Advancing stability analysis of mean-risk stochastic programs: bilevel and two-stage models. Ph.D. Thesis, Universität Dusburg-Essen (2016) – reference: MarinacciMModel uncertaintyJ. Eur. Econ. Assoc.2015131022110010.1111/jeea.12164 – reference: LiuYXuHStability analysis of stochastic programs with second order dominance constraintsMath. Program.2013142435460312708110.1007/s10107-012-0585-0 – reference: FarquharPHUtility assessment methodsManag. Sci.1984301283130077474610.1287/mnsc.30.11.1283 – reference: WeberMDecision making with incomplete informationEur. J. Oper. Res.198728445710.1016/0377-2217(87)90168-8 – reference: Delage, E., Guo, S., Xu, H.: Shortfall risk models when information of loss function is incomplete. Working paper (2017) – reference: HuJMehrotraSRobust decision making over a set of random targets or risk-averse utilities with an application to portfolio optimizationIIE Trans.20154735837210.1080/0740817X.2014.919045 – reference: RockafellarRTWetsRJ-BVariational Analysis1998BerlinSpringer10.1007/978-3-642-02431-3 – reference: GilboaISchmeidlerDMaxmin expected utility with non-unique priorJ. Math. Econ.19891814115310.1016/0304-4068(89)90018-9 – reference: KahnemanDTverskyAProspect Theory: an analysis of decisions under riskEconometrica197947263291361858010.2307/1914185 – reference: AndersonEJBusiness Risk Management: Models and Analysis2014ChichesterWiley – reference: KarmarkarUSSubjectively weighted utility: a descriptive extension of the expected utility modelOrgan. Behav. Hum. 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SubjectTerms	Calculus of Variations and Optimal Control; Optimization Combinatorics Continuity (mathematics) Decision making Estimators Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Optimization models Perturbation Risk management Statistical analysis Theoretical
Title	Statistical robustness in utility preference robust optimization models
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