Advances in Mathematical Modeling, Optimization and Optimal Control PDF
Preview Advances in Mathematical Modeling, Optimization and Optimal Control
Springer Optimization and Its Applications 109 Jean-Baptiste Hiriart-Urruty Adam Korytowski Helmut Maurer Maciej Szymkat Editors Advances in Mathematical Modeling, Optimization and Optimal Control Springer Optimization and Its Applications VOLUME109 ManagingEditor PanosM.Pardalos(UniversityofFlorida) Editor–CombinatorialOptimization Ding-ZhuDu(UniversityofTexasatDallas) AdvisoryBoard J.Birge(UniversityofChicago) C.A.Floudas(TexasA&MUniversity) F.Giannessi(UniversityofPisa) H.D.Sherali(VirginiaPolytechnicandStateUniversity) T.Terlaky(LehighUniversity) Y.Ye(StanfordUniversity) AimsandScope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques havebeendeveloped,thediffusionintootherdisciplineshasproceededata rapidpace,andourknowledgeofallaspectsofthefieldhasgrownevenmore profound.Atthesametime,oneofthemoststrikingtrendsinoptimization is the constantly increasing emphasis on the interdisciplinary nature of the field.Optimizationhasbeenabasictoolinallareasofappliedmathematics, engineering,medicine,economics,andothersciences. The series Springer Optimization and Its Applications publishes under- graduate and graduate textbooks, monographs and state-of-the-art exposi- tory work that focus on algorithms for solving optimization problems and alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objectiveprogramming,descriptionofsoftwarepackages,approxima- tiontechniquesandheuristicapproaches. Moreinformationaboutthisseriesathttp://www.springer.com/series/7393 Jean-Baptiste Hiriart-Urruty • Adam Korytowski Helmut Maurer • Maciej Szymkat Editors Advances in Mathematical Modeling, Optimization and Optimal Control 123 Editors Jean-BaptisteHiriart-Urruty AdamKorytowski InstitutdeMathématiques DepartmentofAutomaticsandBiomedical UniversitéPaulSabatier Engineering Toulouse,France AGHUniversityofScienceandTechnology Kraków,Poland HelmutMaurer InstituteofComputationalandApplied MaciejSzymkat Mathematics DepartmentofAutomaticsandBiomedical UniversityofMünster Engineering Münster,Germany AGHUniversityofScienceandTechnology Kraków,Poland ISSN1931-6828 ISSN1931-6836 (electronic) SpringerOptimizationandItsApplications ISBN978-3-319-30784-8 ISBN978-3-319-30785-5 (eBook) DOI10.1007/978-3-319-30785-5 LibraryofCongressControlNumber:2016939392 MathematicsSubjectClassification:49J15,49J20,49J21,49J30,49N90,26B25,49M29,97N20,47J35 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Contents Introduction ...................................................................... 1 Jean-BaptisteHiriart-Urruty,AdamKorytowski,HelmutMaurer, andMaciejSzymkat Bregman Distances in Inverse Problems and Partial DifferentialEquations........................................................... 3 MartinBurger On Global Attractor for Parabolic Partial Differential InclusionandItsTimeSemidiscretization .................................... 35 PiotrKalita PassiveControlofSingularitiesbyTopologicalOptimization: The Second-Order Mixed Shape Derivatives of Energy FunctionalsforVariationalInequalities....................................... 65 GünterLeugering,JanSokołowski,andAntoniZ˙ochowski Optimal Control for Applications in Medical and RehabilitationTechnology:ChallengesandSolutions....................... 103 KatjaMombaur Second-OrderOptimalityConditionsforBrokenExtremals andBang-BangControls:TheoryandApplications ......................... 147 NikolaiP.OsmolovskiiandHelmutMaurer v Introduction Jean-BaptisteHiriart-Urruty,AdamKorytowski,HelmutMaurer, andMaciejSzymkat Thisbookconstitutesacollectionofdevelopedversionsofplenarypaperspresented (withoneexception)atthe16thFrench–German–PolishConferenceonOptimiza- tion, held in Kraków in 2013. They are authored by researchers of international reputeinthefieldofoptimizationandoptimalcontrol.Thebookincludesanumber of new theoretical results and applications in biomechanics, medical technology, imageprocessing,robotcontrol,etc. Thepurposeofthebookwastogivetheauthorsanopportunitytopresenttheir new results to a wider audience than it was possible at the conference, and in an extended, more comprehensive form. The motivation was that the topics of the articlesarerelatedtoareasoftheoryandapplicationsthatareofmostvividinterest tothescientificcommunity,suchasimageprocessing,partialdifferentialinclusions, shape optimization, optimal control in medical and rehabilitation technology, or sufficientconditionsofoptimality. Thefirstpaper,byMartinBurger,providesanoverviewofrecentdevelopments relatedtoBregmandistances.Approachesininverseproblemsandimageprocessing based on Bregman distances are discussed, which have evolved to a standard tool J.-B.Hiriart-Urruty InstitutdeMathématiques,UniversitéPaulSabatier,31062ToulouseCedex09,France A.Korytowski((cid:2)) DepartmentofAutomaticsandBiomedicalEngineering,AGHUniversityofScienceand Technology,30-059Kraków,Poland e-mail:[email protected] H.Maurer UniversityofMünster,InstituteofComputationalandAppliedMathematics, 48149Münster,Germany M.Szymkat AGHUniversityofScienceandTechnology,DepartmentofAppliedComputerScience, 30-059Kraków,Poland ©SpringerInternationalPublishingSwitzerland2016 1 J.-B.Hiriart-Urrutyetal.(eds.),AdvancesinMathematicalModeling,Optimization andOptimalControl,SpringerOptimizationandItsApplications109, DOI10.1007/978-3-319-30785-5_1 2 J.-B.Hiriart-Urrutyetal. inthesefieldsinthelastdecade.Relatedissuesintheanalysisofnonlinearpartial differentialequationswithavariationalstructurearealsoconsidered. The paper by Piotr Kalita studies the operator version of a first order in time partial differential inclusion and its time discretization by implicit Euler scheme. Thesemidiscretetrajectoriesareprovedtoconvergetothesolutionoftheoriginal problem. It is shown that, as times goes to infinity, all trajectories are attracted towards the so-called global attractors. It is also proved that the semidiscrete attractors converge upper-semicontinuously to the global attractor of the time continuousproblem. InthepaperbyGünterLeugering,JanSokołowski,andAntoniZ˙ochowski,non- smoothshapeoptimizationproblemsforvariationalinequalitiesareconsidered.The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. The topological optimization is used for passive control of singularities of weak solutions. The Hadamard directional differentiability is employed to sensitivity analysis. The topological derivatives of nonsmooth integral shape functionals for variational inequalities are derived. The obtained expressions for derivatives prove useful in numerical optimization for contactproblems. Thenextpaper,byKatjaMombaur,isdevotedtoapplicationsofoptimalcontrol and inverse optimal control in the field of medical and rehabilitation technology, in particular in human movement analysis, therapy and improvement by means of medical devices. Efficient methods for the solution of optimal control and inverse optimalcontrolproblemsarediscussed.Exampleapplicationsofthesemethodsare considered in the development of mobility aids for geriatric patients, the design of exoskeletons, the analysis of running motions with prostheses, the optimal functionalelectricalstimulationofhemiplegicpatients,aswellasstabilityanalysis. The last paper, by Nikolai Osmolovskii and Helmut Maurer, provides a survey on no-gap second-order optimality conditions in the calculus of variations and optimalcontrol,andadiscussionoftheirfurtherdevelopment.Suchconditionsare formulated for discontinuous controls in optimal control problems with endpoint andmixedstate-controlconstraints,andafreecontroltime.Forproblemswiththe control appearing linearly in the Pontryagin function, it is shown that the second- order sufficient condition for the Induced Optimization Problem together with the so-calledstrictbang-bangpropertyensuresecond-ordersufficientconditionsforthe originalcontrolproblem.Thetheoreticalresultsareillustratedbythreeapplications: to optimal control of chemotherapy of HIV, time-optimal control of robots, and controloftheRayleighequation. Bregman Distances in Inverse Problems and Partial Differential Equations MartinBurger Abstract The aim of this paper is to provide an overview of recent development relatedtoBregmandistancesoutsideitsnativeareasofoptimizationandstatistics. WediscussapproachesininverseproblemsandimageprocessingbasedonBregman distances, which have evolved to a standard tool in these fields in the last decade. Moreover,wediscussrelatedissuesintheanalysisandnumericalanalysisofnon- linear partial differential equations with a variational structure. For such problems Bregmandistancesappeartobeofsimilarimportance,butarecurrentlyusedonlyin aquitehiddenfashion.WetrytoworkoutexplicitlytheaspectsrelatedtoBregman distances,whichalsoleadtonovelmathematicalquestionsandmayalsostimulate furtherresearchintheseareas. 1 Introduction Bregmandistancesfor(differentiable)convexfunctionals,originallyintroducedin thestudyofproximalalgorithmsin[11]andnamedin[25],areawell-established concept in continuous and discrete optimization in finite dimension. A classical example is the celebrated Bregman projection algorithm for finding points in the intersectionofaffinesubspaces(cf.,e.g.,[26]).Wereferto[26,53]forintroductory andexhaustiveviewsonBregmandistancesinoptimization. Althoughconvexfunctionalsplayaroleinmanyotherbranchesofmathematics, e.g.,inmanyvariationalproblemsandpartialdifferentialequations,thesuitability of Bregman distances in such fields was hardly investigated for several decades. In mathematical imaging and inverse problems the situation changed with the rediscovery and further development of Bregman iterations as an iterative image restorationtechniqueinthecaseoffrequentlyusedregularizationtechniques such astotalvariation(cf.[50]),whichledtosignificantlyimprovedresultscomparedto standardvariationalmodelsandcouldeliminatesystematicerrorstoacertainextent M.Burger((cid:2)) InstitutfürNumerischeundAngewandteMathematik,WestfälischeWilhelms-Universität (WWU)Münster.Einsteinstr.62,D48149Münster,Germany e-mail:[email protected] ©SpringerInternationalPublishingSwitzerland2016 3 J.-B.Hiriart-Urrutyetal.(eds.),AdvancesinMathematicalModeling,Optimization andOptimalControl,SpringerOptimizationandItsApplications109, DOI10.1007/978-3-319-30785-5_2 4 M.Burger (cf.[9,16]).AnotherkeyobservationincreasingtheinterestinBregmandistancesin thesefieldswasthattheycanbeemployedforerrorestimation,inparticularfornot strictlyconvexandnonsmoothfunctionals(cf.[14]),whichpreventnormestimates. AlthoughtherearemanyobviouslinkstothemainrouteofresearchinBregman distances and related optimization algorithms, there are several peculiar aspects that deserve particular discussion. Besides missing smoothness of the considered functionalsandthefactthatproblemsinimaging,inverseproblemsandpartialdif- ferential equations are naturally formulated in infinite-dimensional Banach spaces suchasthespaceoffunctionsofboundedvariationorSobolevspaces,whichhave only been considered in few instances before, a key point is that the motivation for using Bregman distances in these fields often differs significantly from those in optimization and statistics. In the following we want to provide an overview of suchquestionsandconsequentdevelopments,keepinganeyeonpotentialdirections and questions for future research. We start with a section including definitions, examples, and some general properties of Bregman distances, before we survey aspectsofBregmandistancesininverseproblemsandimagingdevelopedinthelast decade.ThenweproceedtoadiscussionofBregmandistancesinpartialdifferential equations,whichislessexplicitandhencethemaingoalistohighlighthiddenuseof Bregmandistancesandmaketheideamoredirectlyaccessibleforfutureresearch. Finallyweconcludewithasectiononrelatedrecentdevelopments. 2 BregmanDistancesandTheir BasicProperties WestartwithadefinitionofaBregmandistance.Intheremainderofthispaper,let X beaBanachspaceandJ:X→R∪{+∞}beconvexfunctionals.Wefirstrecall thedefinitionofsubdifferential,respectively,subgradients. Definition1. ThesubdifferentialofaconvexfunctionalJisdefinedby ∂J(u)={p∈X∗|J(u)+(cid:6)p,v−u(cid:7)≤J(v)forallv∈X}. (1) Anelementp∈∂J(u)iscalledsubgradient. Having defined a subdifferential we can proceed to the definition of Bregman distances,respectively,generalizedBregmandistancesaccordingto[44]. Definition2. The(generalized)BregmandistancerelatedtoaconvexfunctionalJ withsubgradientpisdefinedby Dp(v,u)=J(v)−J(u)−(cid:6)p,v−u(cid:7), (2) J wherep∈∂J(u).ThesymmetricBregmandistanceisdefinedby Dp,q(u,v)=Dp(v,u)+Dq(u,v)=(cid:6)p−q,u−v(cid:7), (3) J J J
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