Linear and nonlinear optimization:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
[2017]
|
| Series: | International Series in Operations Research & Management Science
Volume 253 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis Klappentext |
| Physical Description: | xxxi, 614 Seiten Illustrationen 23.5 cm x 15.5 cm |
| ISBN: | 9781493970537 |
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CONTENTS
LIST OF FIGURES
LIST OF TABLES
LIST OF EXAMPLES xvii
LIST OF ALGORITHMS xxi
PREFACE xxiii
PART I: LINEAR PROGRAMMING (LP)
1 LP MODELS AND APPLICATIONS 1
1.1 A linear programming problem. 1
1.2 Linear programs and linear functions. 5
1.3 LP models and applications. 7
1.4 Exercises . 22
2 LINEAR EQUATIONS AND INEQUALITIES 29
2.1 Equivalent forms of the LP. 30
2.2 Polyhedral sets. 41
2.3 Exercises . 57
3 THE SIMPLEX ALGORITHM 61
3.1 Preliminaries. 62
3.2 A case where the sufficient condition is also necessary . . 70
3.3 Seeking another BFS . . 71
3.4 Pivoting. 74
3.5 Steps of the Simplex Algorithm. 78
3.6 Exercises . 81
IX
X
LINEAR AND NONLINEAR OPTIMIZATION
4 THE SIMPLEX ALGORITHM CONTINUED 85
4.1 Finding a feasible solution. 85
4.2 Dealing with degeneracy. 94
4.3 Revised Simplex Method. 101
4.4 Exercises . Ill
5 DUALITY AND THE DUAL SIMPLEX ALGORITHM 117
5.1 Dual linear programs. 117
5.2 Introduction to the Dual Simplex Algorithm. 130
5.3 Exercises . 135
6 POSTOPTIMALITY ANALYSES 139
6.1 Changing model dimensions. 141
6.2 Ranging (Sensitivity analysis). 153
6.3 Parametric linear programming. 159
6.4 Shadow prices and their interpretation. 173
6.5 Exercises . 178
7 SOME COMPUTATIONAL CONSIDERATIONS 185
7.1 Problems with explicitly bounded variables. 185
7.2 Constructing a starting (feasible) basis. 197
7.3 Steepest-edge rule for incoming column selection. 204
7.4 Structured linear programs. 208
7.5 Computational complexity of the Simplex Algorithm . . . 224
7.6 Exercises . 228
PART II: NONLINEAR PROGRAMMING (NLP)
8 NLP MODELS AND APPLICATIONS 233
8.1 Nonlinear programming. 233
8.2 Unconstrained nonlinear programs. 237
8.3 Linearly constrained nonlinear programs. 246
8.4 Quadratic programming. 249
8.5 Nonlinearly constrained nonlinear programs. 261
8.6 Exercises . 265
9 UNCONSTRAINED OPTIMIZATION 271
9.1 Generic Optimization Algorithm. 272
9.2 Optimality conditions for univariate minimization. 273
9.3 Finite termination versus convergence of algorithms . . . 276
CONTENTS
XI
9.4 Zero-finding methods. 278
9.5 Univariate minimization. 290
9.6 Optimality conditions for multivariate minimization . . . 299
9.7 Methods for minimizing smooth unconstrained functions . 301
9.8 Steplength algorithms. 305
9.9 Exercises . 311
10 DESCENT METHODS 317
10.1 The Steepest-descent Method. 318
10.2 Newton’s Method. 327
10.3 Quasi-Newton Methods. 346
10.4 The Conjugate-gradient Method . 358
10.5 Exercises . 361
11 OPTIMALITY CONDITIONS 369
11.1 Statement of the problem. 370
11.2 First-order optimality conditions. 371
11.3 Second-order optimality conditions. 385
11.4 Convex programs. 391
11.5 Elementary duality theory for nonlinear programming . . 395
11.6 Exercises . 408
12 PROBLEMS WITH LINEAR CONSTRAINTS 419
12.1 Linear equality constraints. 419
12.2 Methods for computing Z. 431
12.3 Linear inequality constraints. 435
12.4 Active Set Methods . 441
12.5 Special cases . 446
12.6 Exercises . 463
13 PROBLEMS WITH NONLINEAR CONSTRAINTS 469
13.1 Nonlinear equality constraints. 469
13.2 Nonlinear inequality constraints. 477
13.3 Overview of algorithm design. 480
13.4 Penalty-function Methods. 482
13.5 Reduced-gradient and Gradient-projection Methods . . . 489
13.6 Augmented Lagrangian Methods. 491
13.7 Projected Lagrangian Methods. 501
13.8 Sequential Quadratic Programming (SQP) Methods . . . 504
13.9 Exercises . 512
LINEAR AND NONLINEAR OPTIMIZATION
tl
XII
14 INTERIOR-POINT METHODS 517
14.1 Barrier-function methods . 518
14.2 Primal barrier-function method for linear programs. 525
14.3 Primal-Dual barrier function for linear programs. 529
14.4 Exercises . 533
APPENDIX 537
A SOME MATHEMATICAL CONCEPTS 537
A.l Basic terminology and notation from set theory. 537
A.2 Norms and metrics. 540
A.3 Properties of vectors and matrices. 544
A.4 Properties of sets in Rn. 548
A.5 Properties of functions on Rn. 554
A.6 Convex sets. 561
A.7 Properties of convex functions. 566
A. 8 Conjugate duality for convex programs. 574
A.9 Exercises . 582
GLOSSARY OF NOTATION 585
BIBLIOGRAPHY 587
INDEX
603
International Series in Operations Research Management Science ISOR 253
Richard W. Cottle • Mukund N. Thapa
Linear and Nonlinear Optimization
This textbook on Linear and Nonlinear Optimization is intended for graduate and
advanced undergraduate students in operations research and related fields. It is both
literate and mathematically strong, yet requires no prior course in optimization. As
suggested by its title, the book is divided into two parts covering in their individual
chapters LP Models and Applications; Linear Equations and Inequalities; The Simplex
Algorithm; Simplex Algorithm Continued; Duality and the Dual Simplex Algorithm;
Postoptimality Analyses; Computational Considerations; Nonlinear (NLP) Models and
Applications; Unconstrained Optimization; Descent Methods; Optimality Conditions;
Problems with Linear Constraints; Problems with Nonlinear Constraints; Interior-Point
Methods; and an Appendix covering Mathematical Concepts. Each chapter ends with
a set of exercises.
The book is based on lecture notes the authors have used in numerous optimization
courses the authors have taught at Stanford University. It emphasizes modeling and
numerical algorithms for optimization with continuous (not integer) variables.
The discussion presents the underlying theory without always focusing on formal
mathematical proofs (which can be found in cited references). Another feature of this
book is its inclusion of cultural and historical matters, most often appearing among
the footnotes. |
| any_adam_object | 1 |
| author | Cottle, Richard 1934- Thapa, Mukund Narain |
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| dewey-tens | 330 - Economics |
| discipline | Wirtschaftswissenschaften |
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| spelling | Cottle, Richard 1934- Verfasser (DE-588)170020169 aut Linear and nonlinear optimization Richard W. Cottle, Mukund N. Thapa New York, NY Springer [2017] xxxi, 614 Seiten Illustrationen 23.5 cm x 15.5 cm txt rdacontent n rdamedia nc rdacarrier International Series in Operations Research & Management Science Volume 253 Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Lineare Optimierung (DE-588)4035816-1 s Nichtlineare Optimierung (DE-588)4128192-5 s DE-604 Thapa, Mukund Narain Verfasser (DE-588)124788696 aut Springer Science + Business Media LLC (DE-588)1065492340 pbl Erscheint auch als Online-Ausgabe 978-1-4939-7055-1 International Series in Operations Research & Management Science Volume 253 (DE-604)BV011630976 253 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2294fc0096c0484d9a69bf09d2009b51&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030039099&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030039099&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Cottle, Richard 1934- Thapa, Mukund Narain Linear and nonlinear optimization International Series in Operations Research & Management Science Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4128192-5 (DE-588)4123623-3 |
| title | Linear and nonlinear optimization |
| title_auth | Linear and nonlinear optimization |
| title_exact_search | Linear and nonlinear optimization |
| title_full | Linear and nonlinear optimization Richard W. Cottle, Mukund N. Thapa |
| title_fullStr | Linear and nonlinear optimization Richard W. Cottle, Mukund N. Thapa |
| title_full_unstemmed | Linear and nonlinear optimization Richard W. Cottle, Mukund N. Thapa |
| title_short | Linear and nonlinear optimization |
| title_sort | linear and nonlinear optimization |
| topic | Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| topic_facet | Lineare Optimierung Nichtlineare Optimierung Lehrbuch |
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