Modeling and Optimization: Theory and Applications: MOPTA, Bethlehem, PA, USA, August 2016 Selected Contributions PDF
Preview Modeling and Optimization: Theory and Applications: MOPTA, Bethlehem, PA, USA, August 2016 Selected Contributions
Springer Proceedings in Mathematics & Statistics Martin Takáč Tamás Terlaky Editors Modeling and Optimization: Theory and Applications MOPTA, Bethlehem, PA, USA, August 2016 Selected Contributions Springer Proceedings in Mathematics & Statistics Volume 213 SpringerProceedingsinMathematics&Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most excitingareasofmathematicalandstatisticalresearchtoday. Moreinformationaboutthisseriesathttp://www.springer.com/series/10533 Martin Takácˇ • Tamás Terlaky Editors Modeling and Optimization: Theory and Applications MOPTA, Bethlehem, PA, USA, August 2016 Selected Contributions 123 Editors MartinTakácˇ TamásTerlaky IndustrialandSystems IndustrialandSystems EngineeringDepartment EngineeringDepartment LehighUniversity LehighUniversity Bethlehem,PA,USA Bethlehem,PA,USA ISSN2194-1009 ISSN2194-1017 (electronic) SpringerProceedingsinMathematics&Statistics ISBN978-3-319-66615-0 ISBN978-3-319-66616-7 (eBook) DOI10.1007/978-3-319-66616-7 LibraryofCongressControlNumber:2017956711 MathematicsSubjectClassification(2010):49-06,49Mxx,65Kxx,90-06,90Bxx,90Cxx ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This volume contains a selection of papers that were presented at the Modeling and Optimization: Theory and Applications (MOPTA) Conference held at Lehigh University in Bethlehem, Pennsylvania, USA, between August 17 and August 19, 2016. MOPTA 2016 aimed to bring together a diverse group of researchers and practitioners, working on both theoretical and practical aspects of continuous or discrete optimization. The goal was to host presentations on the exciting developments in different areas of optimization and at the same time provide a settingforcloseinteractionamongtheparticipants. ThetopicscoveredatMOPTA2016variedfromalgorithmsforsolvingconvex, combinatorial, nonlinear, and global optimization problems and addressed the application of optimization techniques in finance, electricity systems, healthcare, machinelearning,andotherleadingfields.Theninepaperscontainedinthisvolume representasampleofthesetopicsandapplicationsandillustratethebroaddiversity ofideasdiscussedattheconference.ThefirstpartofthenameMOPTAhighlights therolethatmodelingplaysinthesolutionofanoptimizationproblem,andindeed, some of the papers in this volume illustrate the benefits of effective modeling techniquestiedwiththeoreticalguarantees. The paper by Befekadu et al. considers a stochastic decision problem, with dynamic risk measures, in which multiple risk-averse agents make their decisions tominimizetheirindividualaccumulatedriskcostsoverafinitetimehorizon.The paperbyKampasetal.focusesontheproblemofpackingofgeneral(nonidentical) ellipses in a circle with a minimum radius. Their approach is based on using embedded Lagrange multipliers. The paper by Chopra et al. considers the convex recoloringproblem,i.e.,torecolorthenodesofacoloredgraphusingthesmallest numberofcolorchanges,suchthateachcolorinducesaconnectedsubgraph.They consideredaconvexrecoloringproblemonatreeandproposedacolumngeneration framework that efficiently solves the large-scale convex recoloring problem. The paper by Jadamba and Raciti proposed a variational inequality formulation of a migration model with random data. They assume a simple model of population distributionbasedonutilityfunctiontheory.Incontrasttorecentwork,theyrefined thepreviousmodelbyallowingrandomfluctuationsinthedataoftheproblem. v vi Preface ThepaperbyChoetal.developsafastandreliablecomputationalframeworkfor theinverseproblemofidentifyingvariableparametersingeneralmixedvariational problems. One of the main contributions of their work is a thorough derivation of efficient computation schemes for the evaluation of the gradient and the Hessian of the output least-squares functional both for a discrete and continuous case. The paper by Smirnov and Dmitrieva considers a minimization of the ` -norm of p Dirichlet-typeboundarycontrolsforthe1Dwaveequation. The paper by Liu and Takácˇ proposed a projected mini-batch semi-stochastic gradient descent method. This work improved both the theoretical complexity and practical performance of the general stochastic gradient descent method. They proved a linear convergence under weak strong convexity assumption for minimizing the sum of smooth convex functions subject to a compact polyhedral set, which remains popular across the machine learning community. The paper by Adams and Anjos considered the projection polytope constraints used during optimization of relaxed semidefinite problems. They proposed a bilevel second- orderconeoptimizationapproachtofindthemaximallyviolatedprojectionpolytope constraint according to a particular depth measure and reformulate the bilevel problem as a single-level mixed binary second-order cone optimization problem. ThepaperbyPappdealswithpolynomialoptimizationproblems;especially,itdeals with an approximation of the cone of nonnegative polynomials with the cone of sum-of-squares polynomials. This approximation is polynomial-time solvable for many NP-hard optimization problems using semidefinite optimization. The paper focusesonthenumericalissueofsuchanapproximationscheme. We thank the sponsors of MOPTA 2016, namely, AIMMS, SAS, Gurobi, and SIAM. We also thank the host, Lehigh University, as well as the rest of the organizingcommittee:FrankCurtis,LuisZuluaga,LarrySnyder,TedRalphs,Katya Scheinberg,RobertStorer,AurélieThiele,BorisDefourny,AlexanderStolyar,and EugenePerevalov. Bethlehem,PA,USA MartinTakácˇ Bethlehem,PA,USA TamásTerlaky May2017 Contents StochasticDecisionProblemswithMultipleRisk-AverseAgents .......... 1 Getachew K. Befekadu, Alexander Veremyev, Vladimir Boginski, andEduardoL.Pasiliao OptimalPackingofGeneralEllipsesinaCircle ............................. 23 FrankJ.Kampas,JánosD.Pintér,andIgnacioCastillo ColumnGenerationApproachtotheConvexRecoloringProblem onaTree........................................................................... 39 SunilChopra,ErginErdem,EunseokKim,andSanghoShim AVariationalInequalityFormulationofaMigrationModelwith RandomData..................................................................... 55 BaasansurenJadambaandFabioRaciti IdentificationinMixedVariationalProblemsbyAdjointMethods withApplications................................................................. 65 M.Cho,B.Jadamba,A.A.Khan,A.A.Oberai,andM.Sama MinimizationoftheL -Norm,p(cid:2)1ofDirichlet-TypeBoundary p Controlsforthe1DWaveEquation............................................ 85 IlyaSmirnovandAnastasiaDmitrieva Projected Semi-Stochastic Gradient Descent Method with Mini-BatchSchemeUnderWeakStrongConvexityAssumption .......... 95 JieLiuandMartinTakácˇ ExactSeparationofk-ProjectionPolytopeConstraints ..................... 119 ElspethAdamsandMiguelF.Anjos UnivariatePolynomialOptimizationwithSum-of-SquaresInterpolants.. 143 DávidPapp vii Stochastic Decision Problems with Multiple Risk-Averse Agents GetachewK.Befekadu,AlexanderVeremyev,VladimirBoginski, andEduardoL.Pasiliao Abstract Weconsiderastochasticdecisionproblem,withdynamicriskmeasures, in which multiple risk-averse agents make their decisions to minimize their indi- vidualaccumulatedrisk-costsoverafinite-timehorizon.Specifically,weintroduce multi-structure dynamic risk measures induced from conditional g-expectations, where the latter are associated with the generator functionals of certain BSDEs thatimplicitlytakeintoaccounttherisk-costfunctionalsoftherisk-averseagents. Here, we also assume that the solutions for such BSDEs almost surely satisfy a stochastic viability property w.r.t. a certain given closed convex set. Using a result similar to that of the Arrow–Barankin–Blackwell theorem, we establish the existenceofconsistentoptimaldecisionsfortherisk-averseagents,whenthesetof all Pareto optimal solutions, in the sense of viscosity solutions, for the associated dynamic programming equations is dense in the given closed convex set. Finally, ApreliminaryversionofthispaperwaspresentedattheModelingandOptimization:Theoryand ApplicationsConference,August17–19,2016,Bethlehem,PA,USA. G.K.Befekadu((cid:2)) NRC,AirForceResearchLaboratory&DepartmentofIndustrialSystemEngineering, UniversityofFlorida-REEF,1350N.PoquitoRd,Shalimar,FL32579,USA e-mail:gbefekadu@ufl.edu A.Veremyev DepartmentofIndustrialSystemEngineering,UniversityofFlorida-REEF,1350 N.PoquitoRd,Shalimar,FL32579,USA e-mail:averemyev@ufl.edu V.Boginski IndustrialEngineering&ManagementSystems,UniversityofCentralFlorida,12800Pegasus Dr.,P.O.Box162993,Orlando,FL32816-2993,USA e-mail:[email protected] E.L.Pasiliao MunitionsDirectorate,AirForceResearchLaboratory,101WestEglinBlvd,EglinAFB, FL32542,USA e-mail:[email protected] ©SpringerInternationalPublishingAG2017 1 M.Takácˇ,T.Terlaky(eds.),ModelingandOptimization:TheoryandApplications, SpringerProceedingsinMathematics&Statistics213, DOI10.1007/978-3-319-66616-7_1 2 G.K.Befekaduetal. we comment on the characteristics of acceptable risks w.r.t. some uncertain future outcomes or costs, where results from the dynamic risk analysis are part of the informationusedintherisk-aversedecisioncriteria. Keywords Dynamic programming equation • Forward-backward SDEs • Multiple risk-averse agents • Pareto optimality • Risk-averse decisions • Value functions • Viscositysolutions 1 Introduction (cid:2) (cid:3) Let (cid:2);F;fFtgt(cid:2)0;P be a probability space, and let fBtgt(cid:2)0 be a d-dimensional standard Brownian motion, whose natural filtration, augmented by all P-null sets, isdenotedbyfFtgt(cid:2)0,sothatitsatisfiesthe usualhypotheses(e.g.,see[21]). We considerthefollowingcontrolled-diffusionprocessoveragivenfinite-timehorizon T > 0 (cid:2) (cid:3) (cid:2) (cid:3) dXu(cid:2) Dm t;Xu(cid:2);.u1;u2;(cid:2)(cid:2)(cid:2) ;un/ dtC(cid:3) t;Xu(cid:2);.u1;u2;(cid:2)(cid:2)(cid:2) ;un/ dB; t t t t t t t t t t Xu(cid:2) Dx; 0(cid:3)t(cid:3)T; (1) 0 where – Xu(cid:2) isanRd-valuedcontrolled-diffusionprocess, (cid:3) – ujisaUj-valuedmeasurabledecisionprocess,whichcorrespondstothejthrisk- (cid:3) averseagent(whereUjisanopencompactsetinRmjQ,withjD1;2;:::;n);and, furthermore,u (cid:2) .u1;u2;(cid:2)(cid:2)(cid:2) ;un/isann-tupleof n Ui-valuedmeasurable (cid:3) (cid:3) (cid:3) (cid:3) iD1 decisionprocessessuchthatforallt>s,.B (cid:4)B /isindependentofu forr (cid:3)s t s r (nonanticipativitycondition)and Z t E ju(cid:4)j2d(cid:4) <1 8t(cid:5)s; s Q – mW Œ0;T(cid:5) (cid:6) Rd (cid:6) n Ui ! Rd is uniformly Lipschitz, with bounded first iD1 derivative,and Q – (cid:3)W Œ0;T(cid:5)(cid:6)Rd(cid:6) niD1Ui !Rd(cid:4)d isLipschitzwiththeleQasteigenvalueof(cid:3)(cid:3)T uniformlyboundedawayfromzeroforall.x;u/2Rd(cid:6) n Ui andt 2Œ0;T(cid:5), iD1 i.e., Y n (cid:3).t;x;u/(cid:3)T.t;x;u/(cid:7)(cid:6)I ; 8.x;u/2Rd(cid:6) Ui; d(cid:4)d iD1 8t2Œ0;T(cid:5); forsome(cid:6)>0.
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