Optimization:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; Heidelberg ; Dordrecht ; London
Springer
[2013]
|
| Edition: | Second edition |
| Series: | Springer texts in statistics
95 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 529 Seiten Diagramme |
| ISBN: | 9781461458371 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Optimization
Autor: Lange, Kenneth
Jahr: 2013
Contents
Preface to the Second Edition vii
Preface to the First Edition ix
1 Elementary Optimization 1
1.1 Introduction....................................................1
1.2 Univariate Optimization......................................1
1.3 Multivariate Optimization....................................7
1.4 Constrained Optimization....................................10
1.5 Problems..............................................17
2 The Seven C s of Analysis 23
2.1 Introduction....................................................23
2.2 Vector and Matrix Norms ....................................23
2.3 Convergence and Completeness..............................26
2.4 The Topology of Rn............................................30
2.5 Continuous Functions..........................................34
2.6 Semicontinuity ................................................42
2.7 Connectedness..................................................44
2.8 Uniform Convergence..........................................46
2.9 Problems........................................................47
3 The Gauge Integral 53
3.1 Introduction....................................................53
3.2 Gauge Functions and ¿-Fine Partitions......................54
xiìi
xiv Contents
3.3 Definition and Basic Properties of the Integral..............57
3.4 The Fundamental Theorem of Calculus......................62
3.5 More Advanced Topics in Integration........................66
3.6 Problems........................................................71
4 Differentiation 75
4.1 Introduction....................................................75
4.2 Univariate Derivatives ........................................75
4.3 Partial Derivatives ............................................79
4.4 Differentials....................................................81
4.5 Multivariate Mean Value Theorem ..........................88
4.6 Inverse and Implicit Function Theorems....................89
4.7 Differentials of Matrix-Valued Functions....................93
4.8 Problems........................................................98
5 Karush-Kuhn-Tucker Theory 107
5.1 Introduction....................................................107
5.2 The Multiplier Rule............................................108
5.3 Constraint Qualification......................................114
5.4 Taylor-Made Higher-Order Differentials ..........117
5.5 Applications of Second Differentials..........................123
5.6 Problems........................................................128
6 Convexity 137
6.1 Introduction....................................................137
6.2 Convex Sets....................................................138
6.3 Convex Functions..............................................142
6.4 Continuity, Differentiability, and Integrability.......149
6.5 Minimization of Convex Functions............................152
6.6 Moment Inequalities ..........................................159
6.7 Problems............................162
7 Block Relaxation 171
7.1 Introduction..........................171
7.2 Examples of Block Relaxation................................172
7.3 Problems........................................................180
8 The MM Algorithm 185
8.1 Introduction....................................................185
8.2 Philosophy of the MM Algorithm............................186
8.3 Majorization and Minorization................................187
8.4 Allele Frequency Estimation..................................189
8.5 Linear Regression..............................................191
8.6 Bradley-Terry Model of Ranking..............................193
8.7 Linear Logistic Regression..................194
Contents xv
8.8 Geometrie and Signomial Programs.............194
8.9 Poisson Processes.......................197
8.10 Transmission Tomography..................198
8.11 Poisson Multigraphs .....................202
8.12 Problems............................204
9 The EM Algorithm 221
9.1 Introduction..........................221
9.2 Definition of the EM Algorithm...............222
9.3 Missing Data in the Ordinary Sense.............224
9.4 Allele Frequency Estimation.................225
9.5 Clustering by EM ......................226
9.6 Transmission Tomography..................228
9.7 Factor Analysis........................230
9.8 Hidden Markov Chains....................234
9.9 Problems............................236
10 Newton s Method and Scoring 245
10.1 Introduction..........................245
10.2 Newton s Method and Root Finding.............246
10.3 Newton s Method and Optimization.............248
10.4 MM Gradient Algorithm...................250
10.5 Ad Hoc Approximations of d2f(9)..............252
10.6 Scoring and Exponential Families..............254
10.7 The Gauss-Newton Algorithm................257
10.8 Generalized Linear Models..................258
10.9 Accelerated MM........................259
10.10 Problems............................262
11 Conjugate Gradient and Quasi-Newton 273
11.1 Introduction..........................273
11.2 Centers of Spheres and Centers of Ellipsoids........274
11.3 The Conjugate Gradient Algorithm.............275
11.4 Line Search Methods.....................278
11.5 Stopping Criteria.......................280
11.6 Quasi-Newton Methods....................281
11.7 Trust Regions.........................285
11.8 Problems............................286
12 Analysis of Convergence 291
12.1 Introduction..........................291
12.2 Local Convergence ......................292
12.3 Coercive Functions......................297
12.4 Global Convergence of the MM Algorithm.........299
12.5 Global Convergence of Block Relaxation..........302
xvi Contents
12.6 Global Convergence of Gradient Algorithms........303
12.7 Problems............................306
13 Penalty and Barrier Methods 313
13.1 Introduction..........................313
13.2 Rudiments of Barrier and Penalty Methods.........314
13.3 An Adaptive Barrier Method.................318
13.4 Imposition of a Prior in EM Clustering...........325
13.5 Model Selection and the Lasso................327
13.6 Lasso Penalized i Regression................329
13.7 Lasso Penalized i i Regression................330
13.8 Penalized Discriminant Analysis...............333
13.9 Problems............................334
14 Convex Calculus 341
14.1 Introduction..........................341
14.2 Notation............................342
14.3 Fenchel Conjugates......................342
14.4 Subdifferentials........................351
14.5 The Rules of Convex Differentiation.............358
14.6 Spectral Functions ......................365
14.7 A Convex Lagrange Multiplier Rule.............372
14.8 Problems............................375
15 Feasibility and Duality 383
15.1 Introduction..........................383
15.2 Dykstra s Algorithm .....................384
15.3 Contractive Maps.......................389
15.4 Dual Functions ........................393
15.5 Examples of Dual Programs.................396
15.6 Practical Applications of Duality ..............402
15.7 Problems............................406
16 Convex Minimization Algorithms 415
16.1 Introduction..........................415
16.2 Projected Gradient Algorithm................416
16.3 Exact Penalties and Lagrangians ..............421
16.4 Mechanics of Path Following.................426
16.5 Bregman Iteration.......................432
16.6 Split Bregman Iteration ...................436
16.7 Convergence of Bregman Iteration..............439
16.8 Problems............................440
17 The Calculus of Variations 445
17.1 Introduction..........................445
17.2 Classical Problems ......................446
Contents xvii
17.3 Normed Vector Spaces....................448
17.4 Linear Operators and Functionals..............451
17.5 Differentials..........................453
17.6 The Euler-Lagrange Equation................456
17.7 Applications of the Euler-Lagrange Equation........459
17.8 Lagrange s Lacuna......................462
17.9 Variational Problems with Constraints...........464
17.10 Natural Cubic Splines.....................466
17.11 Problems............................467
Appendix: Mathematical Notes 473
A.l Univariate Normal Random Variables............473
A.2 Multivariate Normal Random Vectors............475
A.3 Polyhedral Sets........................477
A.4 BirkhofF s Theorem and Fan s Inequality ..........480
A.5 Singular Value Decomposition................485
A.6 Hadamard SemidifFerentials .................487
A.7 Problems............................497
References 499
Index 519
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Second edition |
| format | Book |
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| spelling | Lange, Kenneth 1946- Verfasser (DE-588)142260576 aut Optimization Kenneth Lange Second edition New York ; Heidelberg ; Dordrecht ; London Springer [2013] © 2013 xvii, 529 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Springer texts in statistics 95 Algoritmen gtt Mathematische programmering gtt Optimaliseren gtt Optimisation mathématique aMathematical optimization Optimierung (DE-588)4043664-0 gnd rswk-swf Optimierung (DE-588)4043664-0 s DE-604 Erscheint auch als Online-Ausgabe 10.1007/978-1-4614-5838-8 978-1-4614-5838-8 Springer texts in statistics 95 (DE-604)BV041299084 95 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026022305&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lange, Kenneth 1946- Optimization Springer texts in statistics Algoritmen gtt Mathematische programmering gtt Optimaliseren gtt Optimisation mathématique aMathematical optimization Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4043664-0 |
| title | Optimization |
| title_auth | Optimization |
| title_exact_search | Optimization |
| title_full | Optimization Kenneth Lange |
| title_fullStr | Optimization Kenneth Lange |
| title_full_unstemmed | Optimization Kenneth Lange |
| title_short | Optimization |
| title_sort | optimization |
| topic | Algoritmen gtt Mathematische programmering gtt Optimaliseren gtt Optimisation mathématique aMathematical optimization Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Algoritmen Mathematische programmering Optimaliseren Optimisation mathématique aMathematical optimization Optimierung |
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