Risk-Averse Stochastic Programming and Distributionally Robust Optimization Via Operator Splitting
This work deals with a broad class of convex optimization problems under uncertainty. The approach is to pose the original problem as one of finding a zero of the sum of two appropriate monotone operators, which is solved by the celebrated Douglas-Rachford splitting method. The resulting algorithm,...
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| Vydáno v: | Set-valued and variational analysis Ročník 29; číslo 4; s. 861 - 891 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.12.2021
Springer Nature B.V Springer |
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| ISSN: | 1877-0533, 1877-0541 |
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| Abstract | This work deals with a broad class of convex optimization problems under uncertainty. The approach is to pose the original problem as one of finding a zero of the sum of two appropriate monotone operators, which is solved by the celebrated Douglas-Rachford splitting method. The resulting algorithm, suitable for risk-averse stochastic programs and distributionally robust optimization with fixed support, separates the random cost mapping from the risk function composing the problem’s objective. Such a separation is exploited to compute iterates by alternating projections onto different convex sets. Scenario subproblems, free from the risk function and thus parallelizable, are projections onto the cost mappings’ epigraphs. The risk function is handled in an independent and dedicated step consisting of evaluating its proximal mapping that, in many important cases, amounts to projecting onto a certain ambiguity set. Variables get updated by straightforward projections on subspaces through independent computations for the various scenarios. The investigated approach enjoys significant flexibility and opens the way to handle, in a single algorithm, several classes of risk measures and ambiguity sets. |
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| AbstractList | This work deals with a broad class of convex optimization problems under uncertainty. The approach is to pose the original problem as one of finding a zero of the sum of two appropriate monotone operators, which is solved by the celebrated Douglas-Rachford splitting method. The resulting algorithm, suitable for risk-averse stochastic programs and distributionally robust optimization with fixed support, separates the random cost mapping from the risk function composing the problem’s objective. Such a separation is exploited to compute iterates by alternating projections onto different convex sets. Scenario subproblems, free from the risk function and thus parallelizable, are projections onto the cost mappings’ epigraphs. The risk function is handled in an independent and dedicated step consisting of evaluating its proximal mapping that, in many important cases, amounts to projecting onto a certain ambiguity set. Variables get updated by straightforward projections on subspaces through independent computations for the various scenarios. The investigated approach enjoys significant flexibility and opens the way to handle, in a single algorithm, several classes of risk measures and ambiguity sets. |
| Author | de Oliveira, Welington |
| Author_xml | – sequence: 1 givenname: Welington surname: de Oliveira fullname: de Oliveira, Welington email: welington.oliveira@mines-paristech.fr organization: MINES ParisTech, PSL – Research University, CMA – Centre de Mathématiques Appliquées |
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| Keywords | Splitting methods 49J53 49J52 90C25 Progressive hedging Distributionally robust optimization 90C15 Multistage stochastic programs ADMM |
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| References_xml | – reference: de OliveiraWSolodovMA doubly stabilized bundle method for nonsmooth convex optimizationMath. Program.20161561125159345919710.1007/s10107-015-0873-6 – reference: ShapiroADentchevaDRuszczyńskiALectures on Stochastic Programming. Modeling and Theory MPS-SIAM Series on Optimization, vol. 92009PhiladelphiaSIAM and MPS10.1137/1.9780898718751 – reference: RockafellarRConvex Analysis19701st edn.PrincetonPrinceton University Press10.1515/9781400873173 – reference: BorweinJMA very complicated proof of the minimax theoremMinimax Theor. Appl.201611212734778941337.46049 – reference: WatsonJPWoodruffDLProgressive hedging innovations for a class of stochastic mixed-integer resource allocation problemsComput. Manag. Sci.201084355370284220910.1007/s10287-010-0125-4 – reference: Eckstein, J.: Splitting method for monotone operators with applications to parallel optimization. Ph.D. thesis, Massachusetts institut of technology (1989) – reference: Glowinski, R., Le Tallec, P.: Augmented lagrangian methods for the solution of variational problems, 45–121 (1987) – reference: Iutzeler, F., Bianchi, P., Ciblat, P., Hachem, W.: Asynchronous distributed optimization using a randomized alternating direction method of multipliers. In: 52nd IEEE conference on decision and control. IEEE (2013) – reference: RockafellarRTUryasevSThe fundamental risk quadrangle in risk management, optimization and statistical estimationSurveys Oper. Res. Manag. Sci.2013181-233533103448 – reference: ShapiroAKleywegtAMinimax analysis of stochastic problemsOptim. Methods Softw.2002173523542194429410.1080/1055678021000034008 – reference: LionsPLMercierBSplitting algorithms for the sum of two nonlinear operatorsSIAM J. Numer. Anal.197916696497955131910.1137/0716071 – reference: BoydSParikhNChuEPeleatoBEcksteinJDistributed optimization and statistical learning via the alternating direction method of multipliersFound. Trends Mach. Learn.201131112210.1561/2200000016 – reference: RockafellarRTWetsRJBScenarios and policy aggregation in optimization under uncertaintyMath. Oper. Res.1991161119147110679310.1287/moor.16.1.119 – reference: BareillesGLaguelYGrishchenkoDIutzelerFMalickJRandomized progressive hedging methods for multi-stage stochastic programmingAnn. Oper. Res.20202952535560418169910.1007/s10479-020-03811-5 – reference: PengZXuYYanMYinWAROck: An algorithmic framework for asynchronous parallel coordinate updatesSIAM J. Sci. Comput.2016385A2851A2879354485310.1137/15M1024950 – reference: RockafellarRTSolving stochastic programming problems with risk measures by progressive hedgingSet-Valued Variation. Anal.2018264759768388194010.1007/s11228-017-0437-4 – reference: GadeDHackebeilGRyanSMWatsonJPWetsRJBWoodruffDLObtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programsMath. Program.201615714767349206710.1007/s10107-016-1000-z – reference: BonnansJGilbertJLemaréchalCSagastizábalCNumerical Optimization: Theoretical and Practical Aspects20062nd edn.BerlinSpringer1108.65060 – reference: LenoirAMaheyPA survey on operator splitting and decomposition of convex programsRAIRO Oper. Res.20175111741358926210.1051/ro/2015065 – reference: Ben-TalAEl GhaouiLNemirovskiARobust Optimization. Princeton series in Applied Mathematics2009PrincetonPrinceton University Press1221.90001 – reference: FuAZhangJBoydSAnderson accelerated douglas–rachford splittingSIAM J. Sci. Comput.2020426A3560A3583417158710.1137/19M1290097 – reference: EcksteinJBertsekasDPOn the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operatorsMath. Program.1992551-3293318116818310.1007/BF01581204 – reference: BolandNChristiansenJDandurandBEberhardALinderothJLuedtkeJOliveiraFCombining progressive hedging with a Frank-Wolfe method to compute lagrangian dual bounds in stochastic mixed-integer programmingSIAM J. Optim.201828213121336379906610.1137/16M1076290 – reference: PflugGCPichlerAA distance for multistage stochastic optimization modelsSIAM J. Optim.2012221123290268210.1137/110825054 – reference: BeltránFde OliveiraWFinardiECApplication of scenario tree reduction via quadratic process to medium-term hydrothermal scheduling problemIEEE Trans. Power Syst.20173264351436110.1109/TPWRS.2017.2658444 – reference: RockafellarRTProgressive decoupling of linkages in optimization and variational inequalities with elicitable convexity or monotonicitySet-Valued Variat. Anal.2019274863893402766210.1007/s11228-018-0496-1 – reference: ParikhNBoydSProximal algorithmsFound. Trends Optim.20141312723910.1561/2400000003 – reference: AlvesMMEcksteinJGeremiaMMeloJGRelative-error inertial-relaxed inexact versions of douglas-rachford and ADMM splitting algorithmsComput. Optim. Appl.2020752389422406459510.1007/s10589-019-00165-y – reference: BauschkeHHCombettesPLConvex Analysis and Monotone Operator Theory in Hilbert Spaces20172nd edn.BerlinSpringer International Publishing10.1007/978-3-319-48311-5 – reference: PflugGCPichlerAMultistage Stochastic Optimization2014BerlinSpringer International Publishing10.1007/978-3-319-08843-3 – reference: Eckstein, J., Watson, J.P., Woodruff, D.L.: Asynchronous projective hedging for stochastic programming. Tech. Rep. 6895, Optimization Online. http://www.optimization-online.org/DB_HTML/2018/10/6895.html (2018) – reference: DouglasJRachfordHHOn the numerical solution of heat conduction problems in two and three space variablesTrans. Am. Math. 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| SubjectTerms | Algorithms Ambiguity Analysis Computational geometry Convexity Mapping Mathematics Mathematics and Statistics Optimization Risk Robustness Splitting Stochastic programming Subspaces |
| Title | Risk-Averse Stochastic Programming and Distributionally Robust Optimization Via Operator Splitting |
| URI | https://link.springer.com/article/10.1007/s11228-021-00600-5 https://www.proquest.com/docview/2617803411 https://hal.science/hal-03330577 |
| Volume | 29 |
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