Fixed point theory, variational analysis, and optimization:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Boca Raton [u.a.]
CRC Press
2014
|
| Schriftenreihe: | A Chapman & Hall book
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XX, 347 S. graph. Darst. |
| ISBN: | 9781482222074 1482222078 |
Internformat
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| 245 | 1 | 0 | |a Fixed point theory, variational analysis, and optimization |c ed. by Saleh A. R. Al-Mezel ... |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2014 | |
| 300 | |a XX, 347 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a A Chapman & Hall book | |
| 650 | 4 | |a Mathematical analysis | |
| 650 | 4 | |a Fixed point theory | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 0 | 7 | |a Fixpunkttheorie |0 (DE-588)4293945-8 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Al-Mezel, Saleh A. R. |e Sonstige |4 oth | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4822-2208-1 |
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Datensatz im Suchindex
| _version_ | 1804152569130385408 |
|---|---|
| adam_text | Contents
η
Preface
xi
List of Figures
xv
List of Tables
xvii
Contributors
xix
I Fixed Point Theory
1
1
Common Fixed Points in Convex Metric Spaces
3
Abdul Rahim Khan and
Hafiz
Fukhar-ud-din
1.1
Introduction
........................... 3
1.2
Preliminaries
........................... 4
1.3
Ishikawa Iterative Scheme
.................... 15
1.4
Multistep Iterative Scheme
................... 24
1.5
One-Step Implicit Iterative Scheme
.............. 32
Bibliography
.............................. 39
2
Fixed Points of Nonlinear Semigroups in Modular Function
Spaces
45
B. A. Bin Dehaish and M. A. Khamsi
2.1
Introduction
........................... 45
2.2
Basic Definitions and Properties
................ 46
2.3
Some Geometric Properties of Modular Function Spaces
... 53
2.4
Some Fixed-Point Theorems in Modular Spaces
....... 59
2.5
Semigroups in Modular Function Spaces
............ 61
2.6
Fixed Points of Semigroup of Mappings
............ 64
Bibliography
.............................. 71
3
Approximation and Selection Methods for Set-Valued Maps
and Fixed Point Theory
77
Hichem Ben-El-Mechmekh
3.1
Introduction
........................... 78
vi
Contents
3.2 Approximative
Neighborhood Retracts, Extensors, and Space
Approximation
.......................... 80
3.2.1
Approximative Neighborhood Retracts and Extensors
80
3.2.2
Contractibility and Connectedness
........... 84
3.2.2.1
Contractible Spaces
.............. 84
3.2.2.2
Proximal Connectedness
........... 85
3.2.3
Convexity Structures
................... 86
3.2.4
Space Approximation
.................. 90
3.2.4.1
The Property A(iC; V) for Spaces
...... 90
3.2.4.2
Domination of Domain
............ 92
3.2.4.3
Domination, Extension, and Approximation
. 95
3.3
Set-Valued Maps, Continuous Selections, and Approximations
97
3.3.1
Semicontinuity Concepts
................. 98
3.3.2
USC
Approachable Maps and Their Properties
.... 99
3.3.2.1
Conservation of Approachability
....... 100
3.3.2.2
Homotopy Approximation, Domination of
Domain, and Approachability
......... 106
3.3.3
Examples of
А
-Maps..................
108
3.3.4
Continuous Selections for LSC Maps
.......... 113
3.3.4.1
Michael Selections
............... 114
3.3.4.2
A Hybrid Continuous Approximation-Selection
Property
.................... 116
3.3.4.3
More on Continuous Selections for Non-
Convex Maps
.................. 116
3.3.4.4
Non-Expansive Selections
........... 121
3.4
Fixed Point and Coincidence Theorems
............ 122
3.4.1
Generalizations of the
Himmelberg
Theorem to the
Non-Convex Setting
................... 122
3.4.1.1
Preservation of the
FPP
from V to
AţJC;
V)
123
3.4.1.2
A Leray-Schauder Alternative for Approach¬
able Maps
................... 126
3.4.2
Coincidence Theorems
.................. 127
Bibliography
.............................. 131
II Convex Analysis and Variational Analysis
137
4
Convexity, Generalized Convexity, and Applications
139
N.
Hadjisawas
4.1
Introduction
............................ 139
4.2
Preliminaries
........................... 140
4.3
Convex Fbinctions
........................ 141
4.4
Quasiconvex Functions
..................... 148
4.5
Pseudoconvex Functions
.................... 157
Contents
vii
4.6
On the Minima of Generalized Convex Functions
....... 161
4.7
Applications
........................... 163
4.7.1
Sufficiency of the KKT Conditions
........... 163
4.7.2
Applications in Economics
................ 164
4.8
Further Reading
..................... ... 166
Bibliography
.............................. 167
5
New Developments in Quasiconvex Optimization
171
D.
Aussei
5.1
Introduction
........................... 171
5.2
Notations
............................. 174
5.3
The Class of Quasiconvex Functions
.............. 176
5.3.1
Continuity Properties of Quasiconvex Functions
.... 181
5.3.2
Differentiability Properties of Quasiconvex Functions
. 182
5.3.3
Associated Monotonicities
................ 183
5.4
Normal Operator: A Natural Tool for Quasiconvex Functions
184
5.4.1
The Semistrictly Quasiconvex Case
........... 185
5.4.2
The Adjusted
Sublevei
Set and Adjusted Normal Oper¬
ator
............................ 188
5.4.2.1
Adjusted Normal Operator: Definitions
... 188
5.4.2.2
Some Properties of the Adjusted Normal
Operator
.................... 191
5.5
Optimality Conditions for Quasiconvex Programming
.... 196
5.6
Stampacchia Variational Inequalities
.............. 199
5.6.1
Existence Results: The Finite Dimensions Case
.... 199
5.6.2
Existence Results: The Infinite Dimensional Case
. . . 201
5.7
Existence Result for Quasiconvex Programming
....... 203
Bibliography
.............................. 204
β
An Introduction to Variational-like Inequalities
207
Qamrul Hasan Ansari
6.1
Introduction
........................... 207
6.2
Formulations of Variational-like Inequalities
.......... 208
6.3
Variational-like Inequalities and Optimization Problems
. . . 212
6.3.1
Invexity
.......................... 212
6.3.2
Relations between Variational-like Inequalities and an
Optimization Problem
.................. 214
6.4
Existence Theory
........................ 218
6.5
Solution Methods
........................ 225
6.5.1
Auxiliary Principle Method
............... 226
6.5.2
Proximal Method
..................... 231
6.6
Appendix
............................. 238
viii Contents
BibUography
.............................. 240
III Vector
Optimization
247
7
Vector
Optimization:
Basic
Concepts and Solution Methods
249
Dinh The Luc and Augusta
Raţiu
7.1
Introduction
........................... 250
7.2
Mathematical Backgrounds
................... 251
7.2.1
Partial Orders
...................... 252
7.2.2
Increasing Sequences
................... 257
7.2.3
Monotone Functions
................... 258
7.2.4
Biggest Weakly Monotone Functions
.......... 259
7.3
Pareto Maximality
........................ 260
7.3.1
Maximality with Respect to Extended Orders
..... 262
7.3.2
Maximality of Sections
.................. 263
7.3.3
Proper Maximality and Weak Maximality
....... 263
7.3.4
Maximal Points of Free Disposal Hulls
......... 266
7.4
Existence
............................. 268
7.4.1
The Main Theorems
................... 268
7.4.2
Generalization to Order-Complete Sets
......... 269
7.4.3
Existence via Monotone Functions
........... 271
7.5
Vector Optimization Problems
................. 273
7.5.1
Scalarization
....................... 274
7.6
Optimality Conditions
...................... 277
7.6.1
Differentiable Problems
................. 277
7.6.2
Lipschitz Continuous Problems
............. 279
7.6.3
Concave Problems
.................... 281
7.7
Solution Methods
........................ 282
7.7.1
Weighting Method
.................... 282
7.7.2
Constraint Method
.................... 292
7.7.3
Outer Approximation Method
.............. 302
Bibliography
.............................. 305
8
Multi-objective Combinatorial Optimization
307
Matthias Ehrgott and
Xavier
Gandiòleux
8.1
Introduction
........................... 307
8.2
Definitions and Properties
................... 308
8.3
Two Easy Problems: Multi-objective Shortest Path and
Spanning Tree
.......................... 313
8.4
Nice Problems: The Two-Phase Method
............ 315
8.4.1
The Two-Phase Method for Two Objectives
...... 315
8.4.2
The Two-Phase Method for Three Objectives
..... 319
Contents ix
8.5
Difficult
Problems: Scalarization
and Branch and Bound
. . 320
8.5.1 Scalarization....................... 321
8.5.2
Multi-objective Branch and Bound
........... 324
8.6
Challenging Problems: Metaheuristics
............. 327
8.7
Conclusion
............................ 333
Bibliography
.............................. 334
Index
343
Mathematics
Fixed Point Theory, Variational Analysis, and Optimization not only covers
three vital branches of nonlinear analysis—fixed point theory, variational
inequalities, and vector optimization —but also explains the connections
between them, enabling the study of a general form of variational inequality
problems related to the optimality conditions involving differentiable or
directionally differentiable functions. This essential reference supplies both
an introduction to the field and a guideline to the literature, progressing
from basic concepts to the latest developments. Packed with detailed
proofs and bibliographies for further reading, the text:
•
Examines Mann-type iterations for nonlinear mappings on some
classes of a metric space
•
Outlines recent research in fixed point theory in modular function
spaces
•
Discusses key results on the existence of continuous approximations
and selections for set-valued maps with an emphasis on the
nonconvex case
•
Contains definitions, properties, and characterizations of convex,
quasiconvex. and pseudoconvex functions, and of their strict
counterparts
•
Discusses variational inequalities and variational-like inequalities and
their applications
•
Gives an introduction to multi-objective optimization and optimality
conditions
•
Explores mufti-objective combinatorial optimization
(MOCO)
problems,
or integer programs with multiple objectives
Fixed Point Theory, Variational Analysis, and Optimization is a beneficial
resource for the research and study of nonlinear analysis, optimization
theory, vanational inequalities, and mathematical economics. It provides
fundamental knowledge of directional derivatives and
monotonicity
required
in understanding and solving variational inequality problems.
|
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| id | DE-604.BV042103785 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:12:48Z |
| institution | BVB |
| isbn | 9781482222074 1482222078 |
| language | English |
| lccn | 014013180 |
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| spelling | Fixed point theory, variational analysis, and optimization ed. by Saleh A. R. Al-Mezel ... Boca Raton [u.a.] CRC Press 2014 XX, 347 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Mathematical analysis Fixed point theory Mathematical optimization Fixpunkttheorie (DE-588)4293945-8 gnd rswk-swf Fixpunkttheorie (DE-588)4293945-8 s DE-604 Al-Mezel, Saleh A. R. Sonstige oth Erscheint auch als Online-Ausgabe 978-1-4822-2208-1 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027544354&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027544354&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Fixed point theory, variational analysis, and optimization Mathematical analysis Fixed point theory Mathematical optimization Fixpunkttheorie (DE-588)4293945-8 gnd |
| subject_GND | (DE-588)4293945-8 |
| title | Fixed point theory, variational analysis, and optimization |
| title_auth | Fixed point theory, variational analysis, and optimization |
| title_exact_search | Fixed point theory, variational analysis, and optimization |
| title_full | Fixed point theory, variational analysis, and optimization ed. by Saleh A. R. Al-Mezel ... |
| title_fullStr | Fixed point theory, variational analysis, and optimization ed. by Saleh A. R. Al-Mezel ... |
| title_full_unstemmed | Fixed point theory, variational analysis, and optimization ed. by Saleh A. R. Al-Mezel ... |
| title_short | Fixed point theory, variational analysis, and optimization |
| title_sort | fixed point theory variational analysis and optimization |
| topic | Mathematical analysis Fixed point theory Mathematical optimization Fixpunkttheorie (DE-588)4293945-8 gnd |
| topic_facet | Mathematical analysis Fixed point theory Mathematical optimization Fixpunkttheorie |
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