Nonsmooth analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer-Verlag
[2007]
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 351 - 361 |
| Physical Description: | XII, 373 Seiten Diagramme 24 cm |
| ISBN: | 9783540713326 3540713328 |
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MARC
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| 245 | 1 | 0 | |a Nonsmooth analysis |c Winfried Schirotzek |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer-Verlag |c [2007] | |
| 264 | 4 | |c © 2007 | |
| 300 | |a XII, 373 Seiten |b Diagramme |c 24 cm | ||
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| 500 | |a Literaturverz. S. 351 - 361 | ||
| 650 | 4 | |a Optimisation non différentiable | |
| 650 | 4 | |a Nonsmooth optimization | |
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Record in the Search Index
| _version_ | 1804137158040092673 |
|---|---|
| adam_text | Contents
Introduction
................................................... 1
1
Preliminaries
.............................................. 5
1.1
Terminology
............................................ 5
1.2
Convex Sets in Normed Vector Spaces
..................... 6
1.3
Convex Functional: Definitions and Examples
.............. 8
1.4
Continuity of Convex Functionals
......................... 11
1.5
Sandwich and Separation Theorems
....................... 13
1.6
Dual Pairs of Vector Spaces
.............................. 19
1.7
Lower Semicontinuous Functionals
........................ 22
1.8
Bibliographical Notes and Exercises
....................... 24
2
The Conjugate of Convex Functionals
..................... 27
2.1
The Gamma Regularization
.............................. 27
2.2
Conjugate Functionals
................................... 29
2.3
A Theorem of
Hörmander
and the Bipolar Theorem
......... 34
2.4
The Generalized
Farkas
Lemma
........................... 36
2.5
Bibliographical Notes and Exercises
....................... 38
3
Classical Derivatives
....................................... 39
3.1
Directional Derivatives
.................................. 39
3.2
First-Order Derivatives
.................................. 41
3.3
Mean Value Theorems
................................... 44
3.4
Relationship between Differentiability Properties
............ 46
3.5
Higher-Order Derivatives
................................ 48
3.6
Some Examples
......................................... 49
3.7
Implicit Function Theorems and Related Results
............ 51
3.8
Bibliographical Notes and Exercises
....................... 57
X
Contents
4
The Subdifferential of Convex Functionals
................. 59
4.1
Definition and First Properties
........................... 59
4.2
Multifonctions:
First Properties
........................... 63
4.3
Subdifferentials,
Fréchet
Derivatives, and
Asplund
Spaces
.... 64
4.4 ·
Subdifferentials and Conjugate Functionals
................. 73
4.5
Further Calculus Rules
.................................. 76
4.6
The Subdifferential of the Norm
.......................... 78
4.7
Differentiable Norms
.................................... 83
4.8
Bibliographical Notes and Exercises
....................... 89
5
Optimality Conditions for Convex Problems
............... 91
5.1
Basic Optimality Conditions
.............................. 91
5.2
Optimali
ty
Under Functional Constraints
.................. 92
5.3
Application to Approximation Theory
..................... 96
5.4
Existence of Minimum Points and the
Ritz
Method
.......... 99
5.5
Application to Boundary Value Problems
..................105
5.6
Bibliographical Notes and Exercises
.......................110
6
Duality of Convex Problems
...............................
Ill
6.1
Duality in Terms of
a
Lagrange
Function
...................
Ill
6.2 Lagrange
Duality and
Gâteaux
Differentiable Functionals
.... 116
6.3
Duality of Boundary Value Problems
......................118
6.4
Duality in Terms of Conjugate Functions
...................122
6.5
Bibliographical Notes and Exercises
.......................129
7
Derivatives and Subdifferentials of Lipschitz Functionals
. . . 131
7.1
Preview: Derivatives and Approximating Cones
.............131
7.2
Upper Convex Approximations
and Locally Convex Functionals
..........................135
7.3
The Subdifferentials of Clarke and Michel-Penot
............139
7.4
Subdifferential Calculus
..................................146
7.5
Bibliographical Notes and Exercises
.......................153
8
Variational Principles
......................................155
8.1
Introduction
............................................155
8.2
The Loewen Wang Variational Principle
...................156
8.3
The Borwein-Preiss Variational Principle
..................161
8.4
The Deville-Godefroy-Zizler Variational Principle
...........162
8.5
Bibliographical Notes and Exercises
.......................166
9
Subdifferentials of Lower Semicontinuous Functionals
......167
9.1
Fréchet
Subdifferentials: First Properties
...................167
9.2
Approximate Sum and Chain Rules
.......................172
9.3
Application to Hamilton-
J
acobi Equations
.................181
9.4
An Approximate Mean Value Theorem
....................182
9.5
Fréchet
Subdifferential vs. Clarke Subdifferential
............184
Contents
XI
9.6 Multidirectional
Mean Value
Theorems....................185
9.7
The Fréchet Subdifferential
of Marginal Functions...........
190
9.8
Bibliographical Notes and Exercises
.......................193
10
Multifonctions
.............................................195
10.1
The Generalized Open Mapping Theorem
..................195
10.2
Systems of Convex Inequalities
...........................197
10.3
Metric Regularity and Linear Openness
....................200
10.4
Openness Bounds of Multifunctions
.......................209
10.5
Weak Metric Regularity and Pseudo-Lipschitz Continuity
.... 211
10.6
Linear Semiopenness and Related Properties
...............213
10.7
Linearly Semiopen Processes
.............................217
10.8
Maximal Monotone Multifunctions
........................219
10.9
Convergence of Sets
.....................................225
10.10
Bibliographical Notes and Exercises
.......................227
11
Tangent and Normal Cones
................................231
11.1
Tangent Cones: First Properties
..........................231
11.2
Normal Cones: First Properties
...........................237
11.3
Tangent and Normal Cones to Epigraphs
..................241
11.4
Representation of Tangent Cones
.........................245
11.5
Contingent Derivatives and a Lyusternik Type Theorem
.....252
11.6
Representation of Normal Cones
..........................255
11.7
Bibliographical Notes and Exercises
.......................261
12
Optirnality Conditions for Nonconvex Problems
...........265
12.1
Basic Optimality Conditions
..............................265
12.2
Application to the Calculus of Variations
..................267
12.3
Multiplier Rules Involving Upper Convex Approximations
.... 272
12.4
Clarke s Multiplier Rule
.................................278
12.5
Approximate Multiplier Rules
............................280
12.6
Bibliographical Notes and Exercises
.......................283
13
Extremal Principles and More Normals
and Subdifferentials
.......................................285
13.1
Mordukhovich Normals and Subdifferentials
................285
13.2
Coderivatives
...........................................294
13.3
Extremal Principles Involving Translations
.................301
13.4
Sequentially Normally Compact Sets
......................309
13.5
Calculus for Mordukhovich Subdifferentials
.................315
13.6
Calculus for Mordukhovich Normals
.......................320
13.7
Optimality Conditions
...................................323
13.8
The Mordukhovich
Subdifferential
of Marginal Functions
.....327
13.9
A Nonsmooth Implicit Function Theorem
..................330
13.10
An Implicit Multifunction Theorem
.......................334
XII Contents
13.11
An Extremal Principle Involving Deformations
..............337
13.12
Application to Alultiobjective Optimization
................340
13.13
Bibliographical Notes and Exercises
.......................343
Appendix: Further Topics
.....................................347
References
.....................................................351
Notation
.......................................................363
Index
..........................................................366
|
| adam_txt |
Contents
Introduction
. 1
1
Preliminaries
. 5
1.1
Terminology
. 5
1.2
Convex Sets in Normed Vector Spaces
. 6
1.3
Convex Functional: Definitions and Examples
. 8
1.4
Continuity of Convex Functionals
. 11
1.5
Sandwich and Separation Theorems
. 13
1.6
Dual Pairs of Vector Spaces
. 19
1.7
Lower Semicontinuous Functionals
. 22
1.8
Bibliographical Notes and Exercises
. 24
2
The Conjugate of Convex Functionals
. 27
2.1
The Gamma Regularization
. 27
2.2
Conjugate Functionals
. 29
2.3
A Theorem of
Hörmander
and the Bipolar Theorem
. 34
2.4
The Generalized
Farkas
Lemma
. 36
2.5
Bibliographical Notes and Exercises
. 38
3
Classical Derivatives
. 39
3.1
Directional Derivatives
. 39
3.2
First-Order Derivatives
. 41
3.3
Mean Value Theorems
. 44
3.4
Relationship between Differentiability Properties
. 46
3.5
Higher-Order Derivatives
. 48
3.6
Some Examples
. 49
3.7
Implicit Function Theorems and Related Results
. 51
3.8
Bibliographical Notes and Exercises
. 57
X
Contents
4
The Subdifferential of Convex Functionals
. 59
4.1
Definition and First Properties
. 59
4.2
Multifonctions:
First Properties
. 63
4.3
Subdifferentials,
Fréchet
Derivatives, and
Asplund
Spaces
. 64
4.4 ·
Subdifferentials and Conjugate Functionals
. 73
4.5
Further Calculus Rules
. 76
4.6
The Subdifferential of the Norm
. 78
4.7
Differentiable Norms
. 83
4.8
Bibliographical Notes and Exercises
. 89
5
Optimality Conditions for Convex Problems
. 91
5.1
Basic Optimality Conditions
. 91
5.2
Optimali
ty
Under Functional Constraints
. 92
5.3
Application to Approximation Theory
. 96
5.4
Existence of Minimum Points and the
Ritz
Method
. 99
5.5
Application to Boundary Value Problems
.105
5.6
Bibliographical Notes and Exercises
.110
6
Duality of Convex Problems
.
Ill
6.1
Duality in Terms of
a
Lagrange
Function
.
Ill
6.2 Lagrange
Duality and
Gâteaux
Differentiable Functionals
. 116
6.3
Duality of Boundary Value Problems
.118
6.4
Duality in Terms of Conjugate Functions
.122
6.5
Bibliographical Notes and Exercises
.129
7
Derivatives and Subdifferentials of Lipschitz Functionals
. . . 131
7.1
Preview: Derivatives and Approximating Cones
.131
7.2
Upper Convex Approximations
and Locally Convex Functionals
.135
7.3
The Subdifferentials of Clarke and Michel-Penot
.139
7.4
Subdifferential Calculus
.146
7.5
Bibliographical Notes and Exercises
.153
8
Variational Principles
.155
8.1
Introduction
.155
8.2
The Loewen Wang Variational Principle
.156
8.3
The Borwein-Preiss Variational Principle
.161
8.4
The Deville-Godefroy-Zizler Variational Principle
.162
8.5
Bibliographical Notes and Exercises
.166
9
Subdifferentials of Lower Semicontinuous Functionals
.167
9.1
Fréchet
Subdifferentials: First Properties
.167
9.2
Approximate Sum and Chain Rules
.172
9.3
Application to Hamilton-
J
acobi Equations
.181
9.4
An Approximate Mean Value Theorem
.182
9.5
Fréchet
Subdifferential vs. Clarke Subdifferential
.184
Contents
XI
9.6 Multidirectional
Mean Value
Theorems.185
9.7
The Fréchet Subdifferential
of Marginal Functions.
190
9.8
Bibliographical Notes and Exercises
.193
10
Multifonctions
.195
10.1
The Generalized Open Mapping Theorem
.195
10.2
Systems of Convex Inequalities
.197
10.3
Metric Regularity and Linear Openness
.200
10.4
Openness Bounds of Multifunctions
.209
10.5
Weak Metric Regularity and Pseudo-Lipschitz Continuity
. 211
10.6
Linear Semiopenness and Related Properties
.213
10.7
Linearly Semiopen Processes
.217
10.8
Maximal Monotone Multifunctions
.219
10.9
Convergence of Sets
.225
10.10
Bibliographical Notes and Exercises
.227
11
Tangent and Normal Cones
.231
11.1
Tangent Cones: First Properties
.231
11.2
Normal Cones: First Properties
.237
11.3
Tangent and Normal Cones to Epigraphs
.241
11.4
Representation of Tangent Cones
.245
11.5
Contingent Derivatives and a Lyusternik Type Theorem
.252
11.6
Representation of Normal Cones
.255
11.7
Bibliographical Notes and Exercises
.261
12
Optirnality Conditions for Nonconvex Problems
.265
12.1
Basic Optimality Conditions
.265
12.2
Application to the Calculus of Variations
.267
12.3
Multiplier Rules Involving Upper Convex Approximations
. 272
12.4
Clarke's Multiplier Rule
.278
12.5
Approximate Multiplier Rules
.280
12.6
Bibliographical Notes and Exercises
.283
13
Extremal Principles and More Normals
and Subdifferentials
.285
13.1
Mordukhovich Normals and Subdifferentials
.285
13.2
Coderivatives
.294
13.3
Extremal Principles Involving Translations
.301
13.4
Sequentially Normally Compact Sets
.309
13.5
Calculus for Mordukhovich Subdifferentials
.315
13.6
Calculus for Mordukhovich Normals
.320
13.7
Optimality Conditions
.323
13.8
The Mordukhovich
Subdifferential
of Marginal Functions
.327
13.9
A Nonsmooth Implicit Function Theorem
.330
13.10
An Implicit Multifunction Theorem
.334
XII Contents
13.11
An Extremal Principle Involving Deformations
.337
13.12
Application to Alultiobjective Optimization
.340
13.13
Bibliographical Notes and Exercises
.343
Appendix: Further Topics
.347
References
.351
Notation
.363
Index
.366 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Schirotzek, Winfried 1939-2021 |
| author_GND | (DE-588)106923021 |
| author_facet | Schirotzek, Winfried 1939-2021 |
| author_role | aut |
| author_sort | Schirotzek, Winfried 1939-2021 |
| author_variant | w s ws |
| building | Verbundindex |
| bvnumber | BV022890099 |
| callnumber-first | Q - Science |
| callnumber-label | QA402 |
| callnumber-raw | QA402.5 |
| callnumber-search | QA402.5 |
| callnumber-sort | QA 3402.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 870 SK 880 SK 660 |
| ctrlnum | (OCoLC)141385288 (DE-599)DNB983129398 |
| dewey-full | 515.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.64 |
| dewey-search | 515.64 |
| dewey-sort | 3515.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022890099 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:53:18Z |
| indexdate | 2024-07-09T21:07:50Z |
| institution | BVB |
| isbn | 9783540713326 3540713328 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016094953 |
| oclc_num | 141385288 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-384 DE-29 DE-11 DE-188 DE-29T DE-355 DE-BY-UBR DE-83 |
| owner_facet | DE-703 DE-20 DE-384 DE-29 DE-11 DE-188 DE-29T DE-355 DE-BY-UBR DE-83 |
| physical | XII, 373 Seiten Diagramme 24 cm |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer-Verlag |
| record_format | marc |
| series2 | Universitext |
| spelling | Schirotzek, Winfried 1939-2021 (DE-588)106923021 aut Nonsmooth analysis Winfried Schirotzek Berlin ; Heidelberg Springer-Verlag [2007] © 2007 XII, 373 Seiten Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier Universitext Literaturverz. S. 351 - 361 Optimisation non différentiable Nonsmooth optimization Nichtglatte Analysis (DE-588)4379207-8 gnd rswk-swf Nichtglatte Analysis (DE-588)4379207-8 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-71333-3 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016094953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schirotzek, Winfried 1939-2021 Nonsmooth analysis Optimisation non différentiable Nonsmooth optimization Nichtglatte Analysis (DE-588)4379207-8 gnd |
| subject_GND | (DE-588)4379207-8 |
| title | Nonsmooth analysis |
| title_auth | Nonsmooth analysis |
| title_exact_search | Nonsmooth analysis |
| title_exact_search_txtP | Nonsmooth analysis |
| title_full | Nonsmooth analysis Winfried Schirotzek |
| title_fullStr | Nonsmooth analysis Winfried Schirotzek |
| title_full_unstemmed | Nonsmooth analysis Winfried Schirotzek |
| title_short | Nonsmooth analysis |
| title_sort | nonsmooth analysis |
| topic | Optimisation non différentiable Nonsmooth optimization Nichtglatte Analysis (DE-588)4379207-8 gnd |
| topic_facet | Optimisation non différentiable Nonsmooth optimization Nichtglatte Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016094953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schirotzekwinfried nonsmoothanalysis |