Approximate Solutions of Common Fixed-Point Problems

457 Pages·2016·3.45 MB·English

by  Alexander J. Zaslavski (auth.)

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
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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