Algorithms for Solving Common Fixed Point Problems PDF
Preview Algorithms for Solving Common Fixed Point Problems
Springer Optimization and Its Applications 132 Alexander J. Zaslavski Algorithms for Solving Common Fixed Point Problems Springer Optimization and Its Applications Volume 132 ManagingEditor PanosM.Pardalos,UniversityofFlorida Editor-CombinatorialOptimization Ding-ZhuDu,UniversityofTexasatDallas AdvisoryBoard J.Birge,UniversityofChicago S.Butenko,TexasA&MUniversity F.Giannessi,UniversityofPisa S.Rebennack,KarlsruheInstituteofTechnology T.Terlaky,LehighUniversity Y.Ye,StanfordUniversity AimsandScope Optimizationhasbeenexpandinginalldirectionsatanastonishingrateduringthe lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped, thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe interdisciplinarynatureofthefield.Optimizationhasbeenabasictoolinallareasof appliedmathematics,engineering,medicine,economicsandothersciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focusonalgorithmsforsolvingoptimizationproblemsandalsostudyapplications involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of soft- warepackages,approximationtechniquesandheuristicapproaches. Moreinformationaboutthisseriesathttp://www.springer.com/series/7393 Alexander J. Zaslavski Algorithms for Solving Common Fixed Point Problems 123 AlexanderJ.Zaslavski DepartmentofMathematics Technion:IsraelInstituteofTechnology Haifa,Israel ISSN1931-6828 ISSN1931-6836 (electronic) SpringerOptimizationandItsApplications ISBN978-3-319-77436-7 ISBN978-3-319-77437-4 (eBook) https://doi.org/10.1007/978-3-319-77437-4 LibraryofCongressControlNumber:2018935405 MathematicsSubjectClassification:47H05,47H09,47H10,47H14 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface In this book, we study approximate solutions of common fixed point and convex feasibilityproblemsinthepresenceofperturbations.Aconvexfeasibilityproblem istofindapointwhichbelongstotheintersectionofagivenfinitefamilyofsubsets ofaHilbertspace.Thisproblemisaspecialcaseofacommonfixedpointproblem which is to find a common fixed point of a finite family of self-mappings of a Hilbert space. The study of these problems has recently been a rapidly growing area of research. This is due not only to theoretical achievements in this area, but also because of numerous applications to engineering and, in particular, to computed tomography and radiation therapy planning. In the book, we consider a number of algorithms, which are known as important tools for solving convex feasibility and common fixed point problems. According to the results known in theliterature,thesealgorithmsshouldconvergetoasolution.Butitisclearthatin practiceitissufficienttofindagoodapproximatesolutioninsteadofconstructinga minimizingsequence.InourrecentbookApproximateSolutionsofCommonFixed Point Problems, Springer, 2016, we analyzed these algorithms and showed that almostallexactiteratesgeneratedbythemareapproximatesolutions.Moreover,we obtainedanestimateofthenumberofiterateswhicharenotapproximatesolutions. This estimate depends on the algorithm but does not depend on the starting point. In this book, our first goal is to generalize these results for perturbed algorithms inthecasewhenperturbationsaresummable.Thesegeneralizations areimportant because such results find interesting applications and are important ingredients in superiorization and perturbation resilience of algorithms. The superiorization methodology works by taking an iterative algorithm, investigating its perturbation resilience, and then using proactively such perturbations in order to “force” the perturbed algorithm to do in addition to its original task something useful. Our second goal is to study approximate solutions of common fixed point problems in thepresenceofperturbationswhicharenotnecessarilysummable.Notethatinour recentbookmentionedearlieritwasshownthatifperturbationsaresmallenough, then we have an approximate solution during a certain number of iterates, and an estimate for this number of iterates was obtained. But these results do not show v vi Preface whathappenswithsubsequentiterates,whenanapproximatedsolutionisobtained. In this book, we show that if our algorithms are cyclic and a computational error is sufficiently small, then beginning from a certain instant of time iterates become approximate solutions. This instant of time depends on the algorithm but does not dependonitsstartingpoint. Thisbookcontainseightchapters.Chapter1isanintroduction.InChapter2,we studyiterativemethodsinmetricspaces.Thedynamicstring-averagingmethodsfor commonfixedpointproblemsinnormedspaceareanalyzedinChapter3.Dynamic stringmethods,forcommonfixedpointproblemsinametricspace,areintroduced and studied in Chapter 4. Chapter 5 is devoted to the study of the convergence of anabstractversionofthealgorithmwhichiscalledintheliteratureascomponent- averagedrowprojectionsorCARP.InChapter6,westudyaproximalalgorithmfor findingacommonzeroofafamilyofmaximalmonotoneoperators.InChapter7, we extend the results of Chapter 6 for a dynamic string-averaging version of the proximal algorithm. In Chapter 8, subgradient projection algorithms for convex feasibilityproblemsarestudiedforinfinite-dimensionalHilbertspaces. Theorems2.1and3.1wereobtainedin [125].Allotherresultsarenew. Haifa,Israel AlexanderJ.Zaslavski November16,2017 Contents 1 Introduction .................................................................. 1 1.1 CommonFixedPointProblemsinaMetricSpace................... 1 1.2 CommonFixedPointProblemsinaHilbertSpace.................. 6 1.3 ProximalPointAlgorithm............................................. 9 1.4 SubgradientProjectionAlgorithms................................... 13 1.5 Examples............................................................... 16 2 IterativeMethodsinMetricSpaces........................................ 19 2.1 TheFirstProblem...................................................... 19 2.2 ProofofTheorem2.1.................................................. 21 2.3 CyclicIterativeMethods .............................................. 25 2.4 CyclicIterativeMethodswithComputationalErrors................ 32 2.5 TheSecondProblem................................................... 35 2.6 ProofofTheorem2.5.................................................. 37 2.7 ProofofTheorem2.6.................................................. 43 2.8 AuxiliaryResults ...................................................... 49 2.9 ProofofTheorem2.7.................................................. 53 2.10 TheThirdProblem..................................................... 53 2.11 ProofofTheorem2.10 ................................................ 56 2.12 ProofofTheorem2.11 ................................................ 61 2.13 ProofofTheorem2.12 ................................................ 66 3 DynamicString-AveragingMethodsinNormedSpaces ................ 69 3.1 Preliminaries........................................................... 69 3.2 TheFirstProblem...................................................... 72 3.3 ProofofTheorem3.1.................................................. 74 3.4 ProofofTheorem3.2.................................................. 85 3.5 ProofofTheorem3.3.................................................. 93 3.6 TheSecondProblem................................................... 95 3.7 ProofofTheorem3.4.................................................. 97 3.8 ProofofTheorem3.5.................................................. 106 3.9 ProofofTheorem3.6.................................................. 118 vii viii Contents 3.10 TheThirdProblem..................................................... 120 3.11 ProofofTheorem3.7.................................................. 122 3.12 ProofofTheorem3.8.................................................. 131 3.13 ProofofTheorem3.9.................................................. 143 4 DynamicString-MaximumMethodsinMetricSpaces ................. 145 4.1 Preliminaries........................................................... 145 4.2 TheFirstProblem...................................................... 147 4.3 ProofofTheorem3.1.................................................. 148 4.4 TheSecondProblem................................................... 157 4.5 ProofofTheorem4.2.................................................. 158 4.6 TheThirdProblem..................................................... 166 4.7 ProofofTheorem4.3.................................................. 167 5 AbstractVersionofCARPAlgorithm .................................... 177 5.1 PreliminariesandMainResults....................................... 177 5.2 AuxiliaryResults ...................................................... 187 5.3 ProofofTheorem5.1.................................................. 189 5.4 AuxiliaryResultsforTheorems5.2,5.3,5.5,and5.6 ............... 199 5.5 ProofofTheorem5.2.................................................. 202 5.6 ProofofTheorem5.3.................................................. 210 5.7 ProofofTheorem5.4.................................................. 211 5.8 ProofofTheorem5.5.................................................. 221 5.9 ProofofTheorem5.6.................................................. 234 6 ProximalPointAlgorithm................................................... 237 6.1 PreliminariesandMainResults....................................... 237 6.2 AuxiliaryResults ...................................................... 242 6.3 ProofofTheorem6.1.................................................. 244 6.4 ProofofTheorem6.2.................................................. 249 6.5 ProofofTheorem6.3.................................................. 253 7 DynamicString-AveragingProximalPointAlgorithm .................. 255 7.1 PreliminariesandMainResults....................................... 255 7.2 ProofofTheorem7.1.................................................. 261 7.3 ProofofTheorem7.2.................................................. 270 7.4 ProofofTheorem7.3.................................................. 278 8 ConvexFeasibilityProblems................................................ 281 8.1 Preliminaries........................................................... 281 8.2 IterativeMethods ...................................................... 282 8.3 AnAuxiliaryResult ................................................... 284 8.4 ProofofTheorem8.3.................................................. 287 8.5 DynamicString-AveragingSubgradientProjectionAlgorithm ..... 292 8.6 ProofofTheorem8.5.................................................. 295 References......................................................................... 307 Index............................................................................... 313 Chapter 1 Introduction In this book we study approximate solutions of common fixed point and convex feasibilityproblemsinthepresenceofperturbations.Aconvexfeasibilityproblem istofindapointwhichbelongstotheintersectionofagivenfinitefamilyofconvex subsetsofaHilbertspace.Thisproblemisaspecialcaseofacommonfixedpoint problem which is to find a common fixed point of a finite family of nonlinear mappings in a Hilbert space. Our goal is to show the convergence of algorithms, which are known as important tools for solving convex feasibility and common fixedpointproblems.Someofthesealgorithmsarediscussedisthischapter. 1.1 CommonFixedPointProblemsin aMetricSpace Let(X,d)beametricspace.Foreachx ∈XandeachnonemptysetE ⊂Xput d(x,E)=inf{d(x,y): y ∈E}. Foreachx ∈Xandeachr >0set B(x,r)={y ∈X : d(x,y)≤r}. Let m be a natural number, c¯ ∈ (0,1) and let P : X → X, i = 1,...,m be i self-mappingsofthespaceX.Supposethatforeveryi ∈{1,...,m}, Fix(P ):={z∈X : P (z)=z}(cid:6)=∅. i i Wealsosupposethatforeveryi ∈{1,...,m}theinequality d(z,x)2 ≥d(z,P (x))2+c¯d(x,P (x))2 i i ©SpringerInternationalPublishingAG,partofSpringerNature2018 1 A.J.Zaslavski,AlgorithmsforSolvingCommonFixedPointProblems,Springer OptimizationandItsApplications132,https://doi.org/10.1007/978-3-319-77437-4_1 2 1 Introduction holdsforallx ∈Xandallz∈Fix(P ).Set i F =∩m Fix(P ). i=1 i ElementsofthesetF aresolutionsofthecommonfixedpointproblem. It should be mentioned that if the space X is Hilbert and for all i = 1,...,m, the mapping P is the projection on a convex closed set C ⊂ X, then we have a i i convexfeasibilityproblemwhichhasnumerousapplicationstoengineeringand,in particular,tocomputedtomographyandradiationtherapyplanning. Forevery(cid:2) >0andeveryi ∈{1,...,m}set F (P )={x ∈X : d(x,P (x))≤(cid:2)}, (cid:2) i i F˜ (P )={y ∈X : d(y,F (P ))≤(cid:2)}, (cid:2) i (cid:2) i F =∩m F (P ), (cid:2) i=1 (cid:2) i F˜ =∩m F˜ (P ). (cid:2) i=1 (cid:2) i ˜ Elements of F (F respectively) are considered as (cid:2)-approximate solutions of (cid:2) (cid:2) thecommonfixedpointproblem. Fixθ ∈XandanaturalnumberN¯ ≥m. DenotebyRthesetofallmappingsr : {1,2,...} → {1,...,m}suchthatfor eachnumberj, {1,...,m}⊂{r(j),...,r(j +N¯ −1)}. Everyr ∈Rgeneratesthefollowingalgorithm. Initialization:selectanarbitraryx ∈X. 0 Iterativestep:givenacurrentiterationpointx calculatethenextiterationpoint k xk+1 by xk+1 =Pr(k+1)(xk). DenotebyCard(A)thecardinalityofasetA.Thefollowingresultispresented inChapter3of [124].Itwasobtainedin [123]. Theorem1.1 LetM >0satisfy B(θ,M)∩F (cid:6)=∅, (cid:2) >0, r ∈R, x ∈B(θ,M) 0
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