Implicit functions and solution mappings: a view from variational analysis
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
[2014]
|
| Edition: | Second edition |
| Series: | Springer series in operations research and financial engineering
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXVIII, 466 Seiten Illustrationen, Diagramme |
| ISBN: | 9781493910366 |
Staff View
MARC
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| 100 | 1 | |a Dončev, Asen L. |d 1948- |e Verfasser |0 (DE-588)110480376 |4 aut | |
| 245 | 1 | 0 | |a Implicit functions and solution mappings |b a view from variational analysis |c Asen L. Dontchev ; R. Tyrrell Rockafellar |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c [2014] | |
| 300 | |a XXVIII, 466 Seiten |b Illustrationen, Diagramme | ||
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Record in the Search Index
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| adam_text | Contents
1
Introduction
and Equation-Solving Background
......................... 1
1.1 [ 1
A J The Classical Inverse Functi on Theorem
....................... 10
1.2
1
1 B j
The Classical Implicit Function Theorem
...................... 19
1.3
Г
1
Cl
Calmness
......................................................... 25
1.4 [ 1 DJ
Lipschitz Continuity
............................................. 29
1.5 [
E J
Lipschitz Invertibility from Approximations
................... 39
1
.6 [ 1
FJ Selections of
Multi-
valued inverses
............................ 51
1.7
[1GJ Selections from Nonstrict Differentiability
..................... 56
1.8
[IH]
An Implicit Function Theorem for Monotone
Functions
........................................................ 61
1.9 [ 1
1] Commentary
..................................................... 65
2
Solution Mappings for Variational Problems
............................. 69
2.1
[2AJ Generalized Equations and Variational Problems
.............. 70
2.2
[2B] Implicit Function Theorems for Generalized Equations
....... 83
2.3
[2CJ Ample Parameterization and Parametric Robustness
.......... 94
2.4
IŢ2DJ
Semidifferenti
able Functions
.................................... 99
2.5
[2E] Variational Inequalities with Polyhedral Convexity
............ 106
2.6
12FJ Variational Inequalities with
Monotonicity
..................... 118
2.7
[2G] Consequences for Optimization
................................. 123
2.8
[2H]
Commentary
..................................................... 141
3
Set-Valued Analysis of Solution Mappings
................................ 143
3.1
[ЗА]
Set Convergence
................................................. 145
3.2 [3
B j
Continuity of Set-Valued Mappings
............................ 154
3.3
[3C3 Lipschitz Continuity of Set-Valued Mappings
.................
1
60
3.4
[3DJ Outer Lipschitz Continuity
...................................... 167
3.5
[3EJ
Aubin
Property, Metric Regularity, and Linear Openness
..... 172
3.6
[3F] Implicit Mapping Theorems with Metric Regularity
........... 183
3.7
[3G] Strong Metric Regularity
........................................ 192
3.8
[3H | Calmness and Metric Subregulurity
.............................
І
97
XV
xvj
Contents
3.9
[ЗІ]
Strong Metric Subregularity
..................................... 201
ЗЛО
[3J] Commentary
..................................................... 212
4
Metric Regularity Through Generalized Derivatives
.................... 213
4.1
[4A] Graphical Differentiation
........................................ 213
4.2
[4B] Graphical Derivative Criterion for Metric Regularity
.......... 221
4.3
[4C] Coderivative Criterion for Metric Regularity
................... 231
4.4
[4D] Strict Derivative Criterion for Strong Metric Regularity
....... 238
4.5
[4E] Derivative Criterion for Strong Metric Subregularity
.......... 245
4.6
[4F] Applications to Parameterized Constraint Systems
............ 248
4.7
[4C] Isolated Calmness for Variational Inequalities
................. 253
4.8
[4H] Variational Inequalities over Polyhedral Convex Sets
......... 257
4.9 [41]
Strong Metric Regularity of the KKT Mapping
................ 265
4.10
[4J] Commentary
..................................................... 276
5
Metric Regularity in Infinite Dimensions
................................. 277
5.
1 [5A] Positively Homogeneous Mappings
............................ 279
5.2
[5B] Mappings with Closed Convex Graphs
......................... 286
5.3
[5C]
Sublinear
Mappings
............................................. 292
5.4
I5DJ The Theorems of Lyusternik and Graves
....................... 302
5.5
[5Ej Extending the Lyusternik—Graves Theorem
.................... 308
5.6
[5F1 Strong Metric Regularity and Implicit Functions
.............. 318
5.7
[5G] Parametric Inverse Function Theorems
......................... 322
5.8
[5Щ
Further Extensions in Metric Spaces
............................ 328
5.9 [51]
Metric Regularity and Fixed Points
............................. 338
5.
1
0
15J] The Bartle-Graves Theorem and Extensions
................... 344
5.11
[5K] Selections from Directional Differentiability
................... 355
5.12
[5LJ Commentary
..................................................... 359
6
Applications in Numerical
Variational
Analysis
.......................... 363
6.1
[6A] Radius Theorems and Conditioning
............................ 364
6.2
[6B] Constraints and Feasibility
...................................... 373
6.3
[6CJ Iterative Processes for Generalized Equations
................. 379
6.4
[6DÌ
Metric Regularity of Newton s Iteration
........................ 388
6.5
[6E] Inexact Newton s Methods under Strong Metric Subregularity
402
6.6
[6F] Nonsmooth Newton s Method
.................................. 411
6.7
i 6G]
Uniform Strong Metric Regularity and Path-Following
....... 425
6.8
16H] Galerkitr
s
Method for Quadratic Minimization
................ 434
6.9
j
61]
Metric Regularity and Optimal Control
........................ 439
6.10
f
6J
]
The Linear-Quadratic Regulator Problem
...................... 443
6.11
[6K] Commentary
..................................................... 454
References
......................................................................... 455
Index
..................................................................... 463
Asen
L.
Dontchev
·
R.
Tyrrell Rockafellar
Implicit Functions and Solution Mappings
A View from Variational Analysis, Second Edition
The implicit function theorem is one of the most important theorems in analysis and
its many variants are basic tools in partial differential equations and numerical analysis.
This second edition of Implicit Functions and Solution Mappings presents an updated
and more complete picture of the field by including solutions of problems that have
been solved since the first edition was published, and places old and new results in a
broader perspective. The purpose of this self-contained work is to provide a reference on
the topic and to provide a unified collection of a number of results which are currently
scattered throughout the literature. Updates to this edition include new sections in
almost all chapters, new exercises and examples, updated commentaries to chapters
and an enlarged index and references section.
From reviews of the first edition:
The book commences with a helpful context-setting preface followed by six chapters.
Each chapter starts with a useful preamble and concludes with a careful and instructive
commentary, while a good set of references, a notation guide, and a somewhat brief
index complete this study.
...
I unreservedly recommended this book to all practitioners
and graduate students interested in modern optimization theory or control theory or to
those just engaged by beautiful analysis cleanly described. (Jonathan Michael Borwein,
IEEE Control Systems Magazine, February,
2012)
This book is devoted to the theory of inverse and implicit functions and some of its
modifications for solution mappings in variational problems.
...
The book is targeted
to a broad audience of researchers, teachers and graduate students. It can be used as
well as a textbook as a reference book on the topic. Undoubtedly, it will be used by
mathematicians dealing with functional and numerical analysis, optimization, adjacent
branches and also by specialists in mechanics, physics, engineering, economics, and so
on. (Peter
Zabreiko, Zentralblatt MATH,
Vol.
1178,2010)
The present monograph will be a most welcome and valuable addition.
...
This book
will save much time and effort, both for those doing research in variational analysis and
for students learning the field. This important contribution fills a gap in the existing
literature. (Stephen M. Robinson, Mathematical Reviews, Issue
2010)
|
| any_adam_object | 1 |
| author | Dončev, Asen L. 1948- Rockafellar, Ralph Tyrrell 1935- |
| author_GND | (DE-588)110480376 (DE-588)115723498 |
| author_facet | Dončev, Asen L. 1948- Rockafellar, Ralph Tyrrell 1935- |
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| author_sort | Dončev, Asen L. 1948- |
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| bvnumber | BV041969410 |
| classification_rvk | SK 420 SK 600 SK 660 |
| classification_tum | MAT 263f MAT 490f |
| ctrlnum | (OCoLC)945365785 (DE-599)BVBBV041969410 |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| isbn | 9781493910366 |
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| spelling | Dončev, Asen L. 1948- Verfasser (DE-588)110480376 aut Implicit functions and solution mappings a view from variational analysis Asen L. Dontchev ; R. Tyrrell Rockafellar Second edition New York, NY [u.a.] Springer [2014] XXVIII, 466 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Springer series in operations research and financial engineering Variationsproblem (DE-588)4187419-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Implizite Funktion (DE-588)4570203-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Implizite Funktion (DE-588)4570203-2 s Variationsproblem (DE-588)4187419-5 s Variationsrechnung (DE-588)4062355-5 s DE-604 Rockafellar, Ralph Tyrrell 1935- Verfasser (DE-588)115723498 aut Erscheint auch als Online-Ausgabe 978-1-4939-1037-3 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027412075&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027412075&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Dončev, Asen L. 1948- Rockafellar, Ralph Tyrrell 1935- Implicit functions and solution mappings a view from variational analysis Variationsproblem (DE-588)4187419-5 gnd Variationsrechnung (DE-588)4062355-5 gnd Implizite Funktion (DE-588)4570203-2 gnd |
| subject_GND | (DE-588)4187419-5 (DE-588)4062355-5 (DE-588)4570203-2 (DE-588)4123623-3 |
| title | Implicit functions and solution mappings a view from variational analysis |
| title_auth | Implicit functions and solution mappings a view from variational analysis |
| title_exact_search | Implicit functions and solution mappings a view from variational analysis |
| title_full | Implicit functions and solution mappings a view from variational analysis Asen L. Dontchev ; R. Tyrrell Rockafellar |
| title_fullStr | Implicit functions and solution mappings a view from variational analysis Asen L. Dontchev ; R. Tyrrell Rockafellar |
| title_full_unstemmed | Implicit functions and solution mappings a view from variational analysis Asen L. Dontchev ; R. Tyrrell Rockafellar |
| title_short | Implicit functions and solution mappings |
| title_sort | implicit functions and solution mappings a view from variational analysis |
| title_sub | a view from variational analysis |
| topic | Variationsproblem (DE-588)4187419-5 gnd Variationsrechnung (DE-588)4062355-5 gnd Implizite Funktion (DE-588)4570203-2 gnd |
| topic_facet | Variationsproblem Variationsrechnung Implizite Funktion Lehrbuch |
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