Lectures on optimization, theory and algorithms:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1978
|
| Schriftenreihe: | Tata Institute of Fundamental Research lectures on mathematics and physics
Mathematics ; 53 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 236 S. |
| ISBN: | 3540088504 0387088504 |
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| 245 | 1 | 0 | |a Lectures on optimization, theory and algorithms |
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| 490 | 1 | |a Tata Institute of Fundamental Research lectures on mathematics and physics : Mathematics |v 53 | |
| 650 | 4 | |a Optimisation mathématique | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Maxima and minima | |
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Datensatz im Suchindex
| _version_ | 1804136126343020544 |
|---|---|
| adam_text | CONTENTS
CHAPTER I DIFFERENTIAL CALCULUS IN NORMED LINEAR SPACES
1. Gateaux derivatives ... 1
2. Taylor s formula ... 7
3. Convexity and Gateaux differentiability ... 11
4. Gateaux differentiability and weak lower semi continuity 14
5. Commutation of derivations
6. Frechet derivatives ... 6
7. Model problem ... 19
CHAPTER II MINIMIZATION OF FUNCTIONALS THEORY 21
1. Minimization without convexity conditions ... 21
2. Minimization with convexity conditions ... 25
3. Applications to the model problem and reduction to
variational inequality ... 30
4. Some functional spaces ... 33
5. Examples ... 36
CHAPTER III MINIMIZATION WITHOUT CONSTRAINTS
ALGORITHMS ... 49
1. Method of descent :(1.1) Generalities ... 51
(1. 2) Convergent choices of the
direction of descent wk... 54
(1.3) Convergent choices of p^ 59
(1.4) Convergence of algorithms 69
2. Generalized Newton s method ... 73
3. Other methods ... 87
iv
CHAPTER IV MINIMIZATION WITH CONSTRAINTS
ALGORITHMS ... 88
1. Linearization method ... 88
2. Centre method ... 105
3. Method of gradient with projection ... 109
4. Minimization in a product space ... 113
(4.1) Statement of the problem ... 113
(4. 2) Minimization with constraints of convex functionals
on products of reflexive Banach spaces ... 116
(4. 3) Main result convergence of the algorithm and
Gauss Seidel method ... 118
(4.4) Some applications differentiable and non
differentiable functionals in finite dimensions 124
(4. 5) Minimization of quadratic functionals on Hilbert
spaces Relaxation method by blocks ... 126
(4.6) Algorithm (of relaxation method) Details ... 128
(4.7) Convergence of the algorithm ... 130
(4. 8) Some examples of relaxation method in finite
dimensional spaces ... 139
(4. 9) Examples in infinite dimensional Hilbert spaces
optimization with constraints in Sobolev spaces 142
CHAPTER V DUALITY AND ITS APPLICATIONS ... 144
1. Preliminaries Recollection of Hahn Banach and
Ky Fan and Sion theorem, Lagrangian and Lagrange
multipliers, Primal and dual problems .. 145
2. Duality in finite dimensional spaces via Hahn Banach
theorem ••• 155
Qualifying hypothesis • • • 159
Some examples of qualifying hypothesis ... 160
I
3. Duality in infinite dimensional spaces via Ky Fan and
Si on theorem ,., 162
(3.1) Duality in the case of a quadratic form ... 163
(3.2) Dual problem ... 170
(3.3) Method of Uzawa ... 173
4, Minimization of non differentiable functionals using
duality examples and algorithm ... 183
VI ELEMENTS OF THEORY OF OPTIMAL CONTROL AND
ELEMENTS OF OPTIMAL DESIGN ... 194
1. Optimal control theory ... 194
(1.1) Formulation of the problem of optimal control 196
(1.2) Duality and existence ... 198
~? (1,3) Elimination of Btate ... 207
(1.4) Approximation ... 211
2. Theory of optimal design ... 215
(2.1) Formulation of the problem of optimal design 217
t (2.2) A simple example ... 221
(2. 3) Computation of the derivative of the cost function 222
(2.4) Hypothesis and results ... 226
BIBLIOGRAPHY ... 233
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|
| adam_txt |
CONTENTS
CHAPTER I DIFFERENTIAL CALCULUS IN NORMED LINEAR SPACES
1. Gateaux derivatives . 1
2. Taylor's formula . 7
3. Convexity and Gateaux differentiability . 11
4. Gateaux differentiability and weak lower semi continuity 14
5. Commutation of derivations
6. Frechet derivatives . 6
7. Model problem . 19
CHAPTER II MINIMIZATION OF FUNCTIONALS THEORY 21
1. Minimization without convexity conditions . 21
2. Minimization with convexity conditions . 25
3. Applications to the model problem and reduction to
variational inequality . 30
4. Some functional spaces . 33
5. Examples . 36
CHAPTER III MINIMIZATION WITHOUT CONSTRAINTS
ALGORITHMS . 49
1. Method of descent :(1.1) Generalities . 51
(1. 2) Convergent choices of the
direction of descent wk. 54
(1.3) Convergent choices of p^ 59
(1.4) Convergence of algorithms 69
2. Generalized Newton's method . 73
3. Other methods . 87
iv
CHAPTER IV MINIMIZATION WITH CONSTRAINTS
ALGORITHMS . 88
1. Linearization method . 88
2. Centre method . 105
3. Method of gradient with projection . 109
4. Minimization in a product space . 113
(4.1) Statement of the problem . 113
(4. 2) Minimization with constraints of convex functionals
on products of reflexive Banach spaces . 116
(4. 3) Main result convergence of the algorithm and
Gauss Seidel method . 118
(4.4) Some applications differentiable and non
differentiable functionals in finite dimensions 124
(4. 5) Minimization of quadratic functionals on Hilbert
spaces Relaxation method by blocks . 126
(4.6) Algorithm (of relaxation method) Details . 128
(4.7) Convergence of the algorithm . 130
(4. 8) Some examples of relaxation method in finite
dimensional spaces . 139
(4. 9) Examples in infinite dimensional Hilbert spaces
optimization with constraints in Sobolev spaces 142
CHAPTER V DUALITY AND ITS APPLICATIONS . 144
1. Preliminaries Recollection of Hahn Banach and
Ky Fan and Sion theorem, Lagrangian and Lagrange
multipliers, Primal and dual problems . 145
2. Duality in finite dimensional spaces via Hahn Banach
theorem ••• 155
Qualifying hypothesis • • • 159
Some examples of qualifying hypothesis . 160
I
3. Duality in infinite dimensional spaces via Ky Fan and
Si on theorem ,., 162
(3.1) Duality in the case of a quadratic form . 163
(3.2) Dual problem . 170
(3.3) Method of Uzawa . 173
4, Minimization of non differentiable functionals using
duality examples and algorithm . 183
VI ELEMENTS OF THEORY OF OPTIMAL CONTROL AND
ELEMENTS OF OPTIMAL DESIGN . 194
1. Optimal control theory . 194
(1.1) Formulation of the problem of optimal control 196
(1.2) Duality and existence . 198
~? (1,3) Elimination of Btate . 207
(1.4) Approximation . 211
2. Theory of optimal design . 215
(2.1) Formulation of the problem of optimal design 217
t (2.2) A simple example . 221
(2. 3) Computation of the derivative of the cost function 222
(2.4) Hypothesis and results . 226
BIBLIOGRAPHY . 233
i
i
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:18:15Z |
| indexdate | 2024-07-09T20:51:26Z |
| institution | BVB |
| isbn | 3540088504 0387088504 |
| language | English |
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| series | Tata Institute of Fundamental Research lectures on mathematics and physics |
| series2 | Tata Institute of Fundamental Research lectures on mathematics and physics : Mathematics |
| spelling | Cea, Jean Verfasser aut Lectures on optimization, theory and algorithms Berlin [u.a.] Springer 1978 236 S. txt rdacontent n rdamedia nc rdacarrier Tata Institute of Fundamental Research lectures on mathematics and physics : Mathematics 53 Optimisation mathématique Mathematical optimization Maxima and minima Algorithmus (DE-588)4001183-5 gnd rswk-swf Operations Research (DE-588)4043586-6 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Minimierung (DE-588)4251074-0 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Algorithmus (DE-588)4001183-5 s DE-604 Mathematik (DE-588)4037944-9 s Minimierung (DE-588)4251074-0 s Operations Research (DE-588)4043586-6 s Optimierung (DE-588)4043664-0 s 1\p DE-604 Tata Institute of Fundamental Research lectures on mathematics and physics Mathematics ; 53 (DE-604)BV000015654 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015366655&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Cea, Jean Lectures on optimization, theory and algorithms Tata Institute of Fundamental Research lectures on mathematics and physics Optimisation mathématique Mathematical optimization Maxima and minima Algorithmus (DE-588)4001183-5 gnd Operations Research (DE-588)4043586-6 gnd Optimierung (DE-588)4043664-0 gnd Minimierung (DE-588)4251074-0 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4043586-6 (DE-588)4043664-0 (DE-588)4251074-0 (DE-588)4037944-9 |
| title | Lectures on optimization, theory and algorithms |
| title_auth | Lectures on optimization, theory and algorithms |
| title_exact_search | Lectures on optimization, theory and algorithms |
| title_exact_search_txtP | Lectures on optimization, theory and algorithms |
| title_full | Lectures on optimization, theory and algorithms |
| title_fullStr | Lectures on optimization, theory and algorithms |
| title_full_unstemmed | Lectures on optimization, theory and algorithms |
| title_short | Lectures on optimization, theory and algorithms |
| title_sort | lectures on optimization theory and algorithms |
| topic | Optimisation mathématique Mathematical optimization Maxima and minima Algorithmus (DE-588)4001183-5 gnd Operations Research (DE-588)4043586-6 gnd Optimierung (DE-588)4043664-0 gnd Minimierung (DE-588)4251074-0 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Optimisation mathématique Mathematical optimization Maxima and minima Algorithmus Operations Research Optimierung Minimierung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015366655&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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