Lagrange multiplier approach to variational problems and applications:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia, PA
Society for Industrial and Applied Mathematics
2008
|
| Series: | Advances in design and control
15 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVIII, 341 S. |
| ISBN: | 9780898716498 |
Staff View
MARC
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| 100 | 1 | |a Ito, Kazufumi |d 1954- |e Verfasser |0 (DE-588)134247477 |4 aut | |
| 245 | 1 | 0 | |a Lagrange multiplier approach to variational problems and applications |c Kazufumi Ito ; Karl Kunisch |
| 264 | 1 | |a Philadelphia, PA |b Society for Industrial and Applied Mathematics |c 2008 | |
| 300 | |a XVIII, 341 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advances in design and control |v 15 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Linear complementarity problem | |
| 650 | 4 | |a Variational inequalities (Mathematics) | |
| 650 | 4 | |a Multipliers (Mathematical analysis) | |
| 650 | 4 | |a Lagrangian functions | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 0 | 7 | |a Lagrange-Multiplikator |0 (DE-588)4401279-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineares Variationsproblem |0 (DE-588)4234622-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lagrange-Multiplikator |0 (DE-588)4401279-2 |D s |
| 689 | 0 | 1 | |a Nichtlineares Variationsproblem |0 (DE-588)4234622-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Kunisch, Karl |d 1952- |e Verfasser |0 (DE-588)128929286 |4 aut | |
| 830 | 0 | |a Advances in design and control |v 15 |w (DE-604)BV021715022 |9 15 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016453197 | ||
Record in the Search Index
| _version_ | 1804137580385533952 |
|---|---|
| adam_text | Contents
Preface
xi
1
Existence
of
Lagrange
Multipliers
1
1.1 Problem Statement
and generalities
...................... 1
1.2
A generalized open mapping theorem
..................... 3
1.3
Regularity and existence of
Lagrange
multipliers
.............. 5
1.4
Applications
.................................. 8
1.5
Weakly singular problems
........................... 17
1.6
Approximation, penalty, and adapted penalty techniques
........... 23
1.6.1
Approximation techniques
....................... 23
1.6.2
Penalty techniques
........................... 24
2
Sensitivity Analysis
27
2.1
Generalities
.................................. 27
2.2
Implicit function theorem
........................... 31
2.3
Stability results
................................ 34
2.4
Lipschitz continuity
.............................. 45
2.5
Differentiability
................................ 53
2.6
Application to optimal control of an ordinary differential equation
..... 62
3
First Order Augmented Lagrangians for Equality and
Finite Rank Inequality Constraints
65
3.1
Generalities
.................................. 65
3.2
Augmentability and sufficient optimality
................... 67
3.3
The first order augmented Lagrangian algorithm
............... 75
3.4
Convergence of Algorithm
ALM....................... 78
3.5
Application to a parameter estimation problem
................ 82
4
Augmented Lagrangian Methods for Nonsmooth, Convex Optimization
87
4.1
Introduction
.................................. 87
4.2
Convex analysis
................................ 89
4.2.1
Conjugate and biconjugate functionals
................ 92
4.2.2
Subdifferential
............................. 95
4.3 Fenchel
duality theory
............................. 98
vii
viii Contents
4.4
Generalized Yosida-Moreau approximation
................. 104
4.5
Optimality systems
.............................. 109
4.6
Augmented Lagrangian method
........................ 114
4.7
Applications
.................................. 119
4.7.1
Binghamflow
............................. 120
4.7.2
Image restoration
........................... 121
4.7.3
Elastoplastic problem
......................... 122
4.7.4
Obstacle problem
........................... 122
4.7.5
Signorini
problem
........................... 124
4.7.6
Friction problem
............................ 125
4.7.7
¿ -fitting
................................ 126
4.7.8
Control problem
............................ 126
5
Newton and SQP Methods
129
5.1
Preliminaries
................................. 129
5.2
Newton method
................................ 133
5.3
SQP and reduced SQP methods
........................ 137
5.4
Optimal control of the Navier-Stokes equations
............... 143
5.4.1
Necessary optimality condition
.................... 145
5.4.2
Sufficient optimality condition
.................... 147
5.4.3
Newton s method for
(5.4.1) ..................... 147
5.5
Newton method for the weakly singular case
................. 148
6
Augmented Lagrangian-SQP Methods
155
6.1
Generalities
.................................. 155
6.2
Equality-constrained problems
........................ 156
6.3
Partial elimination of constraints
....................... 165
6.4
Applications
.................................. 172
6.4.1
An introductory example
....................... 172
6.4.2
A class of nonlinear elliptic optimal control problems
........ 174
6.5
Approximation and mesh-independence
................... 183
6.6
Comments
................................... 186
7
The Primal-Dual Active Set Method
189
7.1
Introduction and basic properties
.......................189
7.2
Monotone class
................................196
7.3
Cone sum preserving class
..........................197
7.4
Diagonally dominated class
..........................200
7.5
Bilateral constraints, diagonally dominated class
..............202
7.6
Nonlinear control problems with bilateral constraints
............206
8
Semismooth Newton Methods I
215
8.1
Introduction
..................................215
8.2
Semismooth functions in finite dimensions
..................217
8.2.1
Basic concepts and the semismooth Newton algorithm
........217
8.2.2
Globalization
.............................222
Contents ¡x
8.2.3
Descent directions
...........................225
8.2.4
A Gauss-Newton algorithm
......................228
8.2.5
A nonlinear complementarity problem
................231
8.3
Semismooth functions in infinite-dimensional spaces
............234
8.4
The primal-dual active set method as a semismooth Newton method
.... 240
8.5
Semismooth Newton methods for a class of nonlinear complementarity
problems
....................................243
8.6
Semismooth Newton methods and regularization
..............246
9
Semismooth Newton Methods II: Applications
253
9.1
В
V-based image restoration problems
....................254
9.2
Friction and contact problems in elasticity
..................263
9.2.1
Generalities
..............................263
9.2.2
Contact problem with
Tresca
friction
.................265
9.2.3
Contact problem with Coulomb friction
................272
10
Parabolic Variational Inequalities
277
10.1
Strong solutions
................................281
10.2
Regularity
...................................291
10.3
Continuity of
q
->■
y(q)
Є
ΖΛ(Ω)......................
292
10.4
Difference schemes and weak solutions
...................297
10.5
Monotone property
..............................302
11
Shape Optimization
305
11.1
Problem statement and generalities
......................305
11.2
Shape derivative
................................308
11.3
Examples
...................................314
11.3.1
Elliptic Dirichlet boundary value problem
..............314
11.3.2
Inverse interface problem
.......................316
11.3.3
Elliptic systems
............................321
11.3.4
Navier-Stokes system
.........................323
Bibliography
327
Index
339
|
| adam_txt |
Contents
Preface
xi
1
Existence
of
Lagrange
Multipliers
1
1.1 Problem Statement
and generalities
. 1
1.2
A generalized open mapping theorem
. 3
1.3
Regularity and existence of
Lagrange
multipliers
. 5
1.4
Applications
. 8
1.5
Weakly singular problems
. 17
1.6
Approximation, penalty, and adapted penalty techniques
. 23
1.6.1
Approximation techniques
. 23
1.6.2
Penalty techniques
. 24
2
Sensitivity Analysis
27
2.1
Generalities
. 27
2.2
Implicit function theorem
. 31
2.3
Stability results
. 34
2.4
Lipschitz continuity
. 45
2.5
Differentiability
. 53
2.6
Application to optimal control of an ordinary differential equation
. 62
3
First Order Augmented Lagrangians for Equality and
Finite Rank Inequality Constraints
65
3.1
Generalities
. 65
3.2
Augmentability and sufficient optimality
. 67
3.3
The first order augmented Lagrangian algorithm
. 75
3.4
Convergence of Algorithm
ALM. 78
3.5
Application to a parameter estimation problem
. 82
4
Augmented Lagrangian Methods for Nonsmooth, Convex Optimization
87
4.1
Introduction
. 87
4.2
Convex analysis
. 89
4.2.1
Conjugate and biconjugate functionals
. 92
4.2.2
Subdifferential
. 95
4.3 Fenchel
duality theory
. 98
vii
viii Contents
4.4
Generalized Yosida-Moreau approximation
. 104
4.5
Optimality systems
. 109
4.6
Augmented Lagrangian method
. 114
4.7
Applications
. 119
4.7.1
Binghamflow
. 120
4.7.2
Image restoration
. 121
4.7.3
Elastoplastic problem
. 122
4.7.4
Obstacle problem
. 122
4.7.5
Signorini
problem
. 124
4.7.6
Friction problem
. 125
4.7.7
¿'-fitting
. 126
4.7.8
Control problem
. 126
5
Newton and SQP Methods
129
5.1
Preliminaries
. 129
5.2
Newton method
. 133
5.3
SQP and reduced SQP methods
. 137
5.4
Optimal control of the Navier-Stokes equations
. 143
5.4.1
Necessary optimality condition
. 145
5.4.2
Sufficient optimality condition
. 147
5.4.3
Newton's method for
(5.4.1) . 147
5.5
Newton method for the weakly singular case
. 148
6
Augmented Lagrangian-SQP Methods
155
6.1
Generalities
. 155
6.2
Equality-constrained problems
. 156
6.3
Partial elimination of constraints
. 165
6.4
Applications
. 172
6.4.1
An introductory example
. 172
6.4.2
A class of nonlinear elliptic optimal control problems
. 174
6.5
Approximation and mesh-independence
. 183
6.6
Comments
. 186
7
The Primal-Dual Active Set Method
189
7.1
Introduction and basic properties
.189
7.2
Monotone class
.196
7.3
Cone sum preserving class
.197
7.4
Diagonally dominated class
.200
7.5
Bilateral constraints, diagonally dominated class
.202
7.6
Nonlinear control problems with bilateral constraints
.206
8
Semismooth Newton Methods I
215
8.1
Introduction
.215
8.2
Semismooth functions in finite dimensions
.217
8.2.1
Basic concepts and the semismooth Newton algorithm
.217
8.2.2
Globalization
.222
Contents ¡x
8.2.3
Descent directions
.225
8.2.4
A Gauss-Newton algorithm
.228
8.2.5
A nonlinear complementarity problem
.231
8.3
Semismooth functions in infinite-dimensional spaces
.234
8.4
The primal-dual active set method as a semismooth Newton method
. 240
8.5
Semismooth Newton methods for a class of nonlinear complementarity
problems
.243
8.6
Semismooth Newton methods and regularization
.246
9
Semismooth Newton Methods II: Applications
253
9.1
В
V-based image restoration problems
.254
9.2
Friction and contact problems in elasticity
.263
9.2.1
Generalities
.263
9.2.2
Contact problem with
Tresca
friction
.265
9.2.3
Contact problem with Coulomb friction
.272
10
Parabolic Variational Inequalities
277
10.1
Strong solutions
.281
10.2
Regularity
.291
10.3
Continuity of
q
->■
y(q)
Є
ΖΛ(Ω).
292
10.4
Difference schemes and weak solutions
.297
10.5
Monotone property
.302
11
Shape Optimization
305
11.1
Problem statement and generalities
.305
11.2
Shape derivative
.308
11.3
Examples
.314
11.3.1
Elliptic Dirichlet boundary value problem
.314
11.3.2
Inverse interface problem
.316
11.3.3
Elliptic systems
.321
11.3.4
Navier-Stokes system
.323
Bibliography
327
Index
339 |
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| author | Ito, Kazufumi 1954- Kunisch, Karl 1952- |
| author_GND | (DE-588)134247477 (DE-588)128929286 |
| author_facet | Ito, Kazufumi 1954- Kunisch, Karl 1952- |
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| author_sort | Ito, Kazufumi 1954- |
| author_variant | k i ki k k kk |
| building | Verbundindex |
| bvnumber | BV023268137 |
| callnumber-first | Q - Science |
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| classification_rvk | SK 660 SK 870 SK 880 |
| classification_tum | MAT 490f |
| ctrlnum | (OCoLC)212204609 (DE-599)BVBBV023268137 |
| dewey-full | 519.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.3 |
| dewey-search | 519.3 |
| dewey-sort | 3519.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023268137 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:34:47Z |
| indexdate | 2024-07-09T21:14:33Z |
| institution | BVB |
| isbn | 9780898716498 |
| language | English |
| lccn | 2008061103 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016453197 |
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| physical | XVIII, 341 S. |
| publishDate | 2008 |
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| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| series | Advances in design and control |
| series2 | Advances in design and control |
| spelling | Ito, Kazufumi 1954- Verfasser (DE-588)134247477 aut Lagrange multiplier approach to variational problems and applications Kazufumi Ito ; Karl Kunisch Philadelphia, PA Society for Industrial and Applied Mathematics 2008 XVIII, 341 S. txt rdacontent n rdamedia nc rdacarrier Advances in design and control 15 Includes bibliographical references and index Linear complementarity problem Variational inequalities (Mathematics) Multipliers (Mathematical analysis) Lagrangian functions Mathematical optimization Lagrange-Multiplikator (DE-588)4401279-2 gnd rswk-swf Nichtlineares Variationsproblem (DE-588)4234622-8 gnd rswk-swf Lagrange-Multiplikator (DE-588)4401279-2 s Nichtlineares Variationsproblem (DE-588)4234622-8 s DE-604 Kunisch, Karl 1952- Verfasser (DE-588)128929286 aut Advances in design and control 15 (DE-604)BV021715022 15 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016453197&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ito, Kazufumi 1954- Kunisch, Karl 1952- Lagrange multiplier approach to variational problems and applications Advances in design and control Linear complementarity problem Variational inequalities (Mathematics) Multipliers (Mathematical analysis) Lagrangian functions Mathematical optimization Lagrange-Multiplikator (DE-588)4401279-2 gnd Nichtlineares Variationsproblem (DE-588)4234622-8 gnd |
| subject_GND | (DE-588)4401279-2 (DE-588)4234622-8 |
| title | Lagrange multiplier approach to variational problems and applications |
| title_auth | Lagrange multiplier approach to variational problems and applications |
| title_exact_search | Lagrange multiplier approach to variational problems and applications |
| title_exact_search_txtP | Lagrange multiplier approach to variational problems and applications |
| title_full | Lagrange multiplier approach to variational problems and applications Kazufumi Ito ; Karl Kunisch |
| title_fullStr | Lagrange multiplier approach to variational problems and applications Kazufumi Ito ; Karl Kunisch |
| title_full_unstemmed | Lagrange multiplier approach to variational problems and applications Kazufumi Ito ; Karl Kunisch |
| title_short | Lagrange multiplier approach to variational problems and applications |
| title_sort | lagrange multiplier approach to variational problems and applications |
| topic | Linear complementarity problem Variational inequalities (Mathematics) Multipliers (Mathematical analysis) Lagrangian functions Mathematical optimization Lagrange-Multiplikator (DE-588)4401279-2 gnd Nichtlineares Variationsproblem (DE-588)4234622-8 gnd |
| topic_facet | Linear complementarity problem Variational inequalities (Mathematics) Multipliers (Mathematical analysis) Lagrangian functions Mathematical optimization Lagrange-Multiplikator Nichtlineares Variationsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016453197&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV021715022 |
| work_keys_str_mv | AT itokazufumi lagrangemultiplierapproachtovariationalproblemsandapplications AT kunischkarl lagrangemultiplierapproachtovariationalproblemsandapplications |