Variational methods in optimization:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Englewood Cliffs, NJ
Prentice-Hall
1974
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 378 S. |
| ISBN: | 0139406271 |
Staff View
MARC
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| 245 | 1 | 0 | |a Variational methods in optimization |c Donald R. Smith |
| 264 | 1 | |a Englewood Cliffs, NJ |b Prentice-Hall |c 1974 | |
| 300 | |a XV, 378 S. | ||
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| 650 | 4 | |a Calcul des variations | |
| 650 | 4 | |a Optimisation mathématique | |
| 650 | 4 | |a Calculus of variations | |
| 650 | 4 | |a Mathematical optimization | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface vii
1. Functional s 1
1.1. Introduction; Examples of Optimization Problems 1
1.2. Vector Spaces 6
1.3. Functionals 9
1.4. Normed Vector Spaces 18
1.5. Continuous Functionals 23
1.6. Linear Functionals 28
2. A Fundamental Necessary Condition
for an Extremum 31
2.1. Introduction 31
2.2. A Fundamental Necessary Condition for an Extremum 33
2.3. Some Remarks on the Gateaux Variation 38
2.4. Examples on the Calculation of Gateaux Variations 41
2.5. An Optimization Problem in Production Planning 48
2.6. Some Remarks on the Frechet Differential 55
xiii
xiv Contents
3. The Euler Lagrange Necessary Condition
for an Extremum with Constraints 58
3.1. Extremum Problems with a Single Constraint 58
3.2. Weak Continuity of Variations 60
3.3. Statement of the Euler Lagrange Multiplier Theorem
for a Single Constraint 62
3.4. Three Examples, and Some Remarks on the Geometrical
Significance of the Multiplier Theorem 64
3.5. Proof of the Euler Lagrange Multiplier Theorem 72
3.6. The Euler Lagrange Multiplier Theorem for Many
Constraints 77
3.7. An Optimum Consumption Policy with Terminal Savings
Constraint During a Period of Inflation 80
3.8. The Meaning of the Euler Lagrange Multipliers 89
3.9. Chaplygin s Problem, or a Modern Version of Queen
Dido s Problem 95
3.10. The John Multiplier Theorem 108
4. Applications of the Euler Lagrange Multiplier Theorem
in the Calculus of Variations 113
4.1. Problems with Fixed End Points 114
4.2. John Bernoulli s Brachistochrone Problem, and
Brachistochrones Through the Earth 126
4.3. Geodesic Curves 138
4.4. Problems with Variable End Points 149
4.5. How to Design a Thrilling Chute the Chute 168
4.6. Functionals Involving Several Unknown
Functions 178
4.7. Fermat s Principle in Geometrical Optics 186
4.8. Hamilton s Principle of Stationary Action; an
Example on Small Vibrations 195
4.9. The McShane Blankinship Curtain Rod Problem;
Functionals Involving Higher Order Derivatives 206
4.10. Functionals Involving Several Independent Variables;
the Minimal Surface Problem 217
4.11. The Vibrating String 227
5. Applications of the Euler Lagrange Multiplier Theorem
to Problems with
Global Pointwise Inequality Constraints 233
5.1. Slack Functions and Composite Curves 233
5.2. An Optimum Consumption Policy with Terminal
Savings Constraint Without Extreme Hardship 247
5.3. A Problem in Production Planning with Inequality
Constraints 261
Contents xv
6. Applications of the Euler Lagrange Multiplier
Theorem in Elementary Control Theory 274
6.1. Introduction 274
6.2. A Rocket Control Problem: Minimum Time 277
6.3. A Rocket Control Problem: Minimum Fuel 283
6.4. A More General Control Problem 288
6.5. A Simple Bang Bang Problem 297
6.6. Some Remarks on the Maximum Principle and
Dynamic Programming 307
7. The Variational Description of
Sturm Liouville Eigenvalues 309
7.1. Introduction to Sturm Liouville Problems 310
7.2. The Relation Between the Lowest Eigenvalue and the
Rayleigh Quotient 314
7.3. The Rayleigh Ritz Method for the Lowest Eigenvalue 318
7.4. Higher Eigenvalues and the Rayleigh Quotient 323
7.5. The Courant Minimax Principle 327
7.6. Some Implications of the Courant Minimax Principle 331
7.7. Further Extensions of the Theory 335
7.8. Some General Remarks on the Ritz Method of
Approximate Minimization 338
8. Some Remarks on the Use of the
Second Variation in Extremum Problems 343
8.1. Higher Order Variations 343
8.2. A Necessary Condition Involving the Second
Variation at an Extremum 347
8.3. Sufficient Conditions for a Local Extremum 348
Appendix 352
Al. The Cauchy and Schwarz Inequalities 352
A2. An Example on Normed Vector Spaces 353
A3. An Integral Inequality 355
A4. A Fundamental Lemma of the Calculus of Variations 355
A5. Du Bois Reymond s Derivation of the Euler Lagrange
Equation 357
A6. A Useful Result from Calculus 360
A7. The Construction of a Certain Function 362
A8. The Fundamental Lemma for the Case of Several
Independent Variables 363
A9. The Kinetic Energy for a Certain Model of an Elastic String 364
A10. The Variation of an Initial Value Problem with
Respect to a Parameter 366
Subject Index 369
Author Index 375
|
| any_adam_object | 1 |
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| dewey-search | 515/.64 |
| dewey-sort | 3515 264 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| indexdate | 2024-07-09T15:49:12Z |
| institution | BVB |
| isbn | 0139406271 |
| language | English |
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| physical | XV, 378 S. |
| publishDate | 1974 |
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| publisher | Prentice-Hall |
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| spelling | Smith, Donald R. 1939- Verfasser (DE-588)118160095 aut Variational methods in optimization Donald R. Smith Englewood Cliffs, NJ Prentice-Hall 1974 XV, 378 S. txt rdacontent n rdamedia nc rdacarrier Calcul des variations Optimisation mathématique Calculus of variations Mathematical optimization Optimierung (DE-588)4043664-0 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Optimierung (DE-588)4043664-0 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001763363&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Smith, Donald R. 1939- Variational methods in optimization Calcul des variations Optimisation mathématique Calculus of variations Mathematical optimization Optimierung (DE-588)4043664-0 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4043664-0 (DE-588)4062355-5 |
| title | Variational methods in optimization |
| title_auth | Variational methods in optimization |
| title_exact_search | Variational methods in optimization |
| title_full | Variational methods in optimization Donald R. Smith |
| title_fullStr | Variational methods in optimization Donald R. Smith |
| title_full_unstemmed | Variational methods in optimization Donald R. Smith |
| title_short | Variational methods in optimization |
| title_sort | variational methods in optimization |
| topic | Calcul des variations Optimisation mathématique Calculus of variations Mathematical optimization Optimierung (DE-588)4043664-0 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Calcul des variations Optimisation mathématique Calculus of variations Mathematical optimization Optimierung Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001763363&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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