Springer Optimization and Its Applications 112 Alexander J. Zaslavski Approximate Solutions of Common Fixed- Point Problems Springer Optimization and Its Applications VOLUME112 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 Alexander J. Zaslavski Approximate Solutions of Common Fixed-Point Problems 123 AlexanderJ.Zaslavski DepartmentofMathematics TheTechnion–IsraelInstitute ofTechnology RishonLeZion,Israel ISSN1931-6828 ISSN1931-6836 (electronic) SpringerOptimizationandItsApplications ISBN978-3-319-33253-6 ISBN978-3-319-33255-0 (eBook) DOI10.1007/978-3-319-33255-0 LibraryofCongressControlNumber:2016942566 MathematicsSubjectClassification(2010):47H05,47H09,47H10,65J15 ©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 Preface The book is devoted to the study of approximate solutions of common fixed point problemsandconvexfeasibilityproblemsinthepresenceofcomputationalerrors. A convex feasibility problem seeks to find a point which belongs to the inter- section of a given finite family of subsets of a Hilbert space. This problem is a special case of a common fixed point problem which examines how to find a common fixed point of a finite family of self-mappings of a Hilbert space. The study of these problems has recently become a rapidly growing area of research. This is not only due to theoretical achievements in this area but also because of numerous applications to engineering and, in particular, to computed tomography andradiationtherapyplanning. We present a number of results on the convergence behavior of algorithms, which are known as important tools for solving convex feasibility problems and commonfixedpointproblems.Accordingtotheresultsknownintheliterature,these algorithms should converge to a solution. In this book, we study these algorithms taking into account computational errors which always present in practice. In this case,theconvergencetoasolutiondoesnottakeplace. Moreover,weshowthatouralgorithmsgenerateagoodapproximatesolution,if computationalerrorsareboundedfromabovebyasmallpositiveconstant.Clearly, inpractice,itissufficienttofindagoodapproximatesolutioninsteadofconstructing aminimizingsequence.Ontheotherhand,practice,computationsinducenumerical errors, and if one uses methods in order to solve minimization problems, these methodsusuallyprovideonlyapproximatesolutionsoftheproblems.Ourmaingoal is,foraknowncomputationalerror,tofindoutwhatanapproximatesolutioncanbe obtainedandhowmanyiteratesoneneedsforthis. The monograph contains twelve chapters. Chapter 1 is an introduction. In Chap.2, we study dynamic string-averaging methods for common fixed point problems in a Hilbert space. Its results are a generalization of the results for the convex feasibility problems obtained in our recent paper in the journal Journal of Nonlinear and Convex Analysis. In Chap.3, using iterative methods, we study common fixed point problems in metric spaces. In Chap.4, approximate solutions of these problems are obtained by dynamic string-averaging methods in normed v vi Preface spaces. Dynamic string methods, for common fixed point problems in a metric space,areintroducedandstudiedinChap.5.Commonfixedpointproblems,inthe spaces with distances of the Bregman type, are analyzed in Chap.6. The results of Chaps.3–6 are new. Chapter 7 is devoted to the study of the convergence of an abstract version of the algorithm which is called in the literature as component- averagedrowprojectionsorCARP.InChap.8,whichisbasedonourrecentpaper published in the journal Nonlinear Analysis, we study a proximal algorithm for findingacommonzeroofafamilyofmaximalmonotoneoperators.InChap.9,we extendtheresultsofChap.8foradynamicstring-averagingversionoftheproximal algorithm. The results of Chaps.7 and 9 are new. In Chaps.10–12, subgradient projection algorithms for convex feasibility problems are studied for finite and infinite Hilbert spaces. The results of these chapters concerning iterative methods wereobtainedinourrecentpaperspublishedintheJournalonOptimizationTheory andApplicationsandintheJournalofApproximationTheory,whiletheresultson theirdynamicstring-averagingversionsarenew. RishonLeZion,Israel AlexanderJ.Zaslavski November16,2015 Contents 1 Introduction................................................................. 1 1.1 CommonFixedPointProblemsinaHilbertSpace ............... 1 1.2 ProximalPointAlgorithm.......................................... 4 1.3 SubgradientProjectionAlgorithms ................................ 8 2 DynamicString-AveragingMethodsinHilbertSpaces ................ 13 2.1 PreliminariesandtheMainResult................................. 13 2.2 ProofofTheorem2.1............................................... 18 2.3 AsymptoticBehaviorofInexactIterates........................... 28 2.4 ProofofTheorem2.11.............................................. 33 2.5 AuxiliaryResults ................................................... 36 2.6 AConvergenceResult.............................................. 42 2.7 AsymptoticBehaviorofExactIterates ............................ 44 3 IterativeMethodsinMetricSpaces ...................................... 49 3.1 TheFirstProblem................................................... 49 3.2 ProofofTheorem3.1............................................... 53 3.3 ProofofTheorem3.3............................................... 57 3.4 TheSecondProblem................................................ 59 3.5 ProofofTheorem3.5............................................... 66 3.6 ProofofTheorem3.7............................................... 69 3.7 TheThirdProblem.................................................. 72 3.8 ProofofTheorem3.5............................................... 80 3.9 ProofofTheorem3.16.............................................. 83 3.10 ProofofTheorem3.18.............................................. 88 3.11 ProofofTheorem3.21.............................................. 91 3.12 ProofofTheorem3.22.............................................. 93 3.13 GenericProperties .................................................. 94 4 DynamicString-AveragingMethodsinNormedSpaces ............... 99 4.1 PreliminariesandtheFirstProblem................................ 99 4.2 ProofofTheorem4.1............................................... 105 vii viii Contents 4.3 ProofofTheorem4.3............................................... 112 4.4 TheSecondProblem................................................ 118 4.5 ProofofTheorem4.5............................................... 128 4.6 ProofofTheorem4.6............................................... 136 4.7 ProofofTheorem4.8............................................... 144 5 DynamicString-MaximumMethodsinMetricSpaces................. 153 5.1 PreliminariesandMainResults.................................... 153 5.2 AuxiliaryResults ................................................... 161 5.3 ProofofTheorem5.1............................................... 165 5.4 ProofofTheorem5.2............................................... 170 5.5 ProofofTheorem5.3............................................... 174 5.6 ProofofTheorem5.4............................................... 179 5.7 ProofofTheorem5.5............................................... 183 5.8 ProofofTheorem5.6............................................... 187 5.9 ProofofTheorem5.7............................................... 193 6 SpaceswithGeneralizedDistances....................................... 199 6.1 PreliminariesandMainResults.................................... 199 6.2 AuxiliaryResults ................................................... 205 6.3 ProofofTheorem6.1............................................... 213 6.4 ProofofTheorem6.2............................................... 217 6.5 ProofofTheorem6.3............................................... 222 6.6 ProofofTheorem6.4............................................... 227 6.7 ProofofTheorem6.5............................................... 230 6.8 ProofofTheorem6.6............................................... 237 6.9 ProofofTheorem6.7............................................... 244 7 AbstractVersionofCARPAlgorithm.................................... 251 7.1 PreliminariesandMainResults.................................... 251 7.2 AuxiliaryResults ................................................... 260 7.3 ProofofTheorem7.1............................................... 262 7.4 ProofofTheorem7.2............................................... 265 7.5 ProofofTheorem7.3............................................... 271 7.6 ProofofTheorem7.4............................................... 279 8 ProximalPointAlgorithm................................................. 289 8.1 PreliminariesandMainResults.................................... 289 8.2 AuxiliaryResults ................................................... 298 8.3 ProofofTheorem8.1............................................... 302 8.4 ProofofTheorem8.2............................................... 306 8.5 ProofofTheorem8.3............................................... 309 8.6 ProofofTheorem8.5............................................... 311 8.7 ProofofTheorem8.8............................................... 311 8.8 ProofofTheorem8.9............................................... 313 8.9 ProofofTheorem8.15.............................................. 314 Contents ix 9 DynamicString-AveragingProximalPointAlgorithm................. 319 9.1 PreliminariesandMainResults.................................... 319 9.2 ProofofTheorem9.1............................................... 325 9.3 ProofofTheorem9.2............................................... 335 10 ConvexFeasibilityProblems .............................................. 341 10.1 IterativeMethodsinInfinite-DimensionalSpaces ................ 341 10.2 ProofofTheorem10.3.............................................. 344 10.3 IterativeMethodsinFinite-DimensionalSpaces.................. 346 10.4 AuxiliaryResults ................................................... 349 10.5 ProofofTheorem10.4.............................................. 350 10.6 ProofofTheorem10.5.............................................. 351 10.7 Dynamic String-Averaging Methods inInfinite-DimensionalSpaces..................................... 357 10.8 ProofofTheorem10.11 ............................................ 360 10.9 Dynamic String-Averaging Methods inFinite-DimensionalSpaces ...................................... 366 10.10 ProofofTheorem10.12 ............................................ 367 10.11 Problems in Finite-Dimensional Spaces withComputationalErrors ......................................... 369 10.12 ProofofTheorem10.13 ............................................ 370 10.13 Extensions........................................................... 380 11 IterativeSubgradientProjectionAlgorithm............................. 385 11.1 Preliminaries ........................................................ 385 11.2 TheFirstMainResult............................................... 388 11.3 TheSecondMainResult............................................ 391 11.4 ProofsofLemmas11.3and11.5................................... 393 11.5 ProofsofTheorems11.2and11.4 ................................. 395 11.6 TheThirdMainResult.............................................. 402 11.7 AuxiliaryResultsforTheorem11.7 ............................... 404 11.8 ProofofTheorem11.7.............................................. 406 12 DynamicString-AveragingSubgradientProjectionAlgorithm....... 411 12.1 PreliminariesandtheFirstMainResult ........................... 411 12.2 ProofofTheorem12.1.............................................. 416 12.3 TheSecondMainResult............................................ 427 12.4 ProofofTheorem12.2.............................................. 429 12.5 TheThirdMainResult.............................................. 441 12.6 ProofofTheorem12.3.............................................. 443 References......................................................................... 447 Index............................................................................... 453
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