Linear and Nonlinear Programming:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham, Switzerland
Springer
[2021]
|
| Edition: | Fifth edition |
| Series: | International series in operations research & management science
volume 228 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 609 Seiten Illustrationen |
| ISBN: | 9783030854492 9783030854522 |
Staff View
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| adam_text | Contents 1 Introduction.................................................................................................. 1.1 Optimization...................................................................................... 1.2 Types of Problems....................................... 1.3 Complexity of Problems............................................ 1.4 Iterative Algorithms and Convergence........ .................................. Parti 1 1 2 6 7 Linear Programming 2 Basic Properties of Linear Programs............................... 2.1 Introduction..................................................................................... 2.2 Examples of Linear Programming Problems ... . . ........................... 2.3 Basic Feasible Solutions................................................................... 2.4 The Fundamental Theorem of Linear Programming..................... 2.5 Relations to Convex Geometry................................... 2.6 Farkas’ Lemma and Alternative Systems.............................. 2.7 Summary......................................................................... 2.8 Exercises..................................... ............. 13 13 16 24 26 28 33 34 35 3 Duality and Complementarity................................................................... 3.1 Dual Linear Programs and Interpretations..................................... 3.2 The Duality Theorem....................................................................... 3.3 Geometric and Economic Interpretations....................................... 3.4 Sensitivity and Complementary
Slackness................................... 3.5 Selected Applications of the Duality............................................... 3.6 Max Flow-Min Cut Theorem............................. 3.7 Summary........................................................................................... 3.8 Exercises........................................................................................... 41 41 47 50 52 56 61 67 67 4 The Simplex Method................................................................................... 4.1 Adjacent Basic Feasible Solutions (Extreme Points)..................... 4.2 The Primal Simplex Method ................................... 4.3 The Dual Simplex Method.................................. 77 78 81 88 xi
Contents xii The Simplex Tableau Method.......................................................... The Simplex Method for Transportation Problems ....................... Efficiency Analysis of the Simplex Method................................... Summary............................................................................................ Exercises............................................................................................ 93 101 114 117 118 5 Interior-Point Methods................................................................................ 5.1 Elements of Complexity Theory...................................................... 5.2 *The Simplex Method Is Not Polynomial-Time............................. 5.3 *The Ellipsoid Method..................................................................... 5.4 The Analytic Center.......................................................................... 5.5 The Central Path................................................................................ 5.6 Solution Strategies............................................................................. 5.7 Termination and Initialization.......................................................... 5.8 Summary............................................................................................ 5.9 Exercises................................. 129 131 132 134 137 141 146 154 160 160 6 Conic Linear Programming........................................................................ 6.1 Convex
Cones.................................................................................... 6.2 Conic Linear Programming Problem .............................................. 6.3 Farkas’ Lemma for Conic Linear Programming............................ 6.4 Conic Linear Programming Duality................................................. 6.5 Complementarity and Solution Rank of SDP ................................ 6.6 Interior-Point Algorithms for Conic Linear Programming............ 6.7 Summary............................................................................................ 6.8 Exercises ......................................................................... 165 165 166 172 176 185 190 194 195 4.4 4.5 4.6 4.7 4.8 Part II Unconstrained Problems 7 Basic Properties of Solutions and Algorithms........................................ 7.1 First-Order Necessary Conditions................................................... 7.2 Examples of Unconstrained Problems............................................ 7.3 Second-Order Conditions................................................................. 7.4 Convex and Concave Functions.................................................... 7.5 Minimization and Maximization of Convex Functions................ 7.6 Global Convergence of Descent Algorithms.............................. 7.7 Speed of Convergence..................................................................... 7.8 Summary............................................. 7.9 Exercises....................................................................... 201 202 205 209 212
215 217 225 230 231 8 Basic Descent Methods.................................................... .:............. 8.1 Line Search Algorithms.............................................................. 8.2 The Method of Steepest Descent: First-Order.............. . 8.3 Applications of the Convergence Theory and Preconditioning... 8.4 Accelerated Steepest Descent.............. .......................................... 8.5 Multiplicative Steepest Descent............. 8.6 Newton’s Method: Second-Order.................................... 235 236 252 264 268 271 275
xiii Contents Sequential Quadratic Optimization Methods............. . .................. Coordinate and Stochastic Gradient Descent Methods.................. Summary................................................. Exercises .................................................................. ;...................... 281 287 294 295 9 Conjugate Direction Methods.................................................................... 9.1 Conjugate Directions....................................................................... 9.2 Descent Properties of the Conjugate Direction Method................ 9.3 The Conjugate Gradient Method..................................................... 9.4 The C-G Method as an Optimal Process....................................... 9.5 The Partial Conjugate Gradient Method......................................... 9.6 Extension to Nonquadratic Problems............................................. 9.7 *Parallel Tangents............................................................................. 9.8 Exercises...................................................................................... 301 301 304 307 309 312 315 318 321 10 Quasi-Newton Methods.............................................................................. 10.1 Modified Newton Method................................................................ 10.2 Construction of the Inverse.............................................................. 10.3 Davidon-Fletcher-Powell Method.................................................. 10.4 The Broy den
Family........................................ 10.5 Convergence Properties.................................................................... 10.6 Scaling................................................................................................ 10.7 Memoryless Quasi-Newton Methods............................................. 10.8 *Combination of Steepest Descent and Newton’s Method.......... 10.9 Summary........................................................................................... 10.10 Exercises........................................................................................... 325 326 328 331 334 337 341 346 348 351 352 8.7 8.8 8.9 8.10 Part ΠΙ Constrained Optimization 11 Constrained Optimization Conditions.................................................... 11.1 Constraints and Tangent Plane........................................................ 11.2 First-Order Necessary Conditions (Equality Constraints) ............ 11.3 Equality Constrained Optimization Examples............................... 11.4 Second-Order Conditions (Equality Constraints) ........................ 11.5 Inequality Constraints ................................................. 11.6 Mix-Constrained Optimization Examples...................................... 11.7 Lagrangian Duality and Zero-Order Conditions............................ 11.8 Rules for Constructing the Lagrangian Dual Explicitly................ 11.9 Summary........................................................................................... 11.10
Exercises........................................................................................... 361 361 366 369 376 381 387 390 395 397 398 12 Primal Methods............................................................................................ 12.1 Infeasible Direction and the Steepest Descent Projection Method................................................................................ 406 12.2 Feasible Direction Methods: Sequential Linear Programming ... 12.3 The Gradient Projection Method..................................................... 12.4 Convergence Rate of the Gradient Projection Method.................. 405 412 414 420
Contents xiv 12.5 12.6 12.7 12.8 12.9 12.10 13 429 435 442 445 449 450 Penalty and Barrier Methods....................................................................... 455 Penalty Methods............................................................................... Barrier Methods................................................................................. Lagrange Multipliers in Penalty and Barrier Methods.................. Newton’s Method for the Logarithmic Barrier Optimization....... Newton’s Method for Equality Constrained Optimization........... Conjugate Gradients and Penalty Methods.................................... Penalty Functions and Gradient Projection.................................... Summary........................................................................................... Exercises........................................................................................... 456 460 463 470 473 476 477 481 482 Local Duality and Dual Methods................................................................ 487 488 494 498 503 508 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14 The Reduced Gradient Method........................................................ Convergence Rate of the Reduced Gradient Method..................... Sequential Quadratic Optimization Methods................................. Active Set Methods.......................................................................... Summary...........................................................................................
Exercises........................................................................................... 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Local Duality and the Lagrangian Method.................................... Separable Problems and Their Duals............................................. The Augmented Lagrangian and Interpretation............................ The Augmented Lagrangian Method of Multipliers ..................... The Alternating Direction Method of Multipliers......................... The Multi-Block Extension of the Alternating Direction Method of Multipliers........................................................ 513 *Cutting Plane Methods.................................................................. Exercises........................................................................................... 515 521 15 Primal-Dual Methods...................................................................................... 525 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 A The Standard Problem and Monotone Function............................ A Simple Merit Function.................................................................. Basic Primal-Dual Methods............................................................. Relation to Sequential Quadratic Optimization ............................. Primal-Dual Interior-Point (Barrier) Methods............................... The Monotone Complementarity Problem..................................... Detect Infeasibility in Nonlinear Optimization.........................
Summary............................................................................................ Exercises........................................................................................... 525 529 531 537 542 547 550 553 554 Mathematical Review..................................................................................... 559 A.l A.2 A.3 A.4 A.5 A.6 Sets .................................................................................................... Matrix Notation....................................................... Spaces................................................................................................ Eigenvalues and Quadratic Forms.................................................... Topological Concepts................................................ Functions........................................................................ 559 560 561 562 564 564
Contents В Convex Sets.......................................................................................................... 571 B.l B.2 B.3 B.4 C Basic Definitions............................................................................... Hyperplanes and Polytopes............................................................... Separating and Supporting Hyperplanes......................................... Extreme Points................................................................................... 571 573 575 577 Gaussian Elimination....................................................................................... 579 C.l C.2 D XV The LU Decomposition..................................................................... 579 Pivots.................................................................................................. 582 Basic Network Concepts.................................................................................. 587 Flows in Networks............................................................................ Tree Procedure.................................................................................. Capacitated Networks ...................................................................... 589 589 591 Bibliography.............................................................................................................. 593 Index............................................................................................................................. 607 D.l D.2 D.3
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| adam_txt |
Contents 1 Introduction. 1.1 Optimization. 1.2 Types of Problems. 1.3 Complexity of Problems. 1.4 Iterative Algorithms and Convergence. . Parti 1 1 2 6 7 Linear Programming 2 Basic Properties of Linear Programs. 2.1 Introduction. 2.2 Examples of Linear Programming Problems . . . . 2.3 Basic Feasible Solutions. 2.4 The Fundamental Theorem of Linear Programming. 2.5 Relations to Convex Geometry. 2.6 Farkas’ Lemma and Alternative Systems. 2.7 Summary. 2.8 Exercises. . 13 13 16 24 26 28 33 34 35 3 Duality and Complementarity. 3.1 Dual Linear Programs and Interpretations. 3.2 The Duality Theorem. 3.3 Geometric and Economic Interpretations. 3.4 Sensitivity and Complementary
Slackness. 3.5 Selected Applications of the Duality. 3.6 Max Flow-Min Cut Theorem. 3.7 Summary. 3.8 Exercises. 41 41 47 50 52 56 61 67 67 4 The Simplex Method. 4.1 Adjacent Basic Feasible Solutions (Extreme Points). 4.2 The Primal Simplex Method . 4.3 The Dual Simplex Method. 77 78 81 88 xi
Contents xii The Simplex Tableau Method. The Simplex Method for Transportation Problems . Efficiency Analysis of the Simplex Method. Summary. Exercises. 93 101 114 117 118 5 Interior-Point Methods. 5.1 Elements of Complexity Theory. 5.2 *The Simplex Method Is Not Polynomial-Time. 5.3 *The Ellipsoid Method. 5.4 The Analytic Center. 5.5 The Central Path. 5.6 Solution Strategies. 5.7 Termination and Initialization. 5.8 Summary. 5.9 Exercises. 129 131 132 134 137 141 146 154 160 160 6 Conic Linear Programming. 6.1 Convex
Cones. 6.2 Conic Linear Programming Problem . 6.3 Farkas’ Lemma for Conic Linear Programming. 6.4 Conic Linear Programming Duality. 6.5 Complementarity and Solution Rank of SDP . 6.6 Interior-Point Algorithms for Conic Linear Programming. 6.7 Summary. 6.8 Exercises . 165 165 166 172 176 185 190 194 195 4.4 4.5 4.6 4.7 4.8 Part II Unconstrained Problems 7 Basic Properties of Solutions and Algorithms. 7.1 First-Order Necessary Conditions. 7.2 Examples of Unconstrained Problems. 7.3 Second-Order Conditions. 7.4 Convex and Concave Functions. 7.5 Minimization and Maximization of Convex Functions. 7.6 Global Convergence of Descent Algorithms. 7.7 Speed of Convergence. 7.8 Summary. 7.9 Exercises. 201 202 205 209 212
215 217 225 230 231 8 Basic Descent Methods. .:. 8.1 Line Search Algorithms. 8.2 The Method of Steepest Descent: First-Order. . 8.3 Applications of the Convergence Theory and Preconditioning. 8.4 Accelerated Steepest Descent. . 8.5 Multiplicative Steepest Descent. 8.6 Newton’s Method: Second-Order. 235 236 252 264 268 271 275
xiii Contents Sequential Quadratic Optimization Methods. . . Coordinate and Stochastic Gradient Descent Methods. Summary. Exercises . ;. 281 287 294 295 9 Conjugate Direction Methods. 9.1 Conjugate Directions. 9.2 Descent Properties of the Conjugate Direction Method. 9.3 The Conjugate Gradient Method. 9.4 The C-G Method as an Optimal Process. 9.5 The Partial Conjugate Gradient Method. 9.6 Extension to Nonquadratic Problems. 9.7 *Parallel Tangents. 9.8 Exercises. 301 301 304 307 309 312 315 318 321 10 Quasi-Newton Methods. 10.1 Modified Newton Method. 10.2 Construction of the Inverse. 10.3 Davidon-Fletcher-Powell Method. 10.4 The Broy den
Family. 10.5 Convergence Properties. 10.6 Scaling. 10.7 Memoryless Quasi-Newton Methods. 10.8 *Combination of Steepest Descent and Newton’s Method. 10.9 Summary. 10.10 Exercises. 325 326 328 331 334 337 341 346 348 351 352 8.7 8.8 8.9 8.10 Part ΠΙ Constrained Optimization 11 Constrained Optimization Conditions. 11.1 Constraints and Tangent Plane. 11.2 First-Order Necessary Conditions (Equality Constraints) . 11.3 Equality Constrained Optimization Examples. 11.4 Second-Order Conditions (Equality Constraints) . 11.5 Inequality Constraints . 11.6 Mix-Constrained Optimization Examples. 11.7 Lagrangian Duality and Zero-Order Conditions. 11.8 Rules for Constructing the Lagrangian Dual Explicitly. 11.9 Summary. 11.10
Exercises. 361 361 366 369 376 381 387 390 395 397 398 12 Primal Methods. 12.1 Infeasible Direction and the Steepest Descent Projection Method. 406 12.2 Feasible Direction Methods: Sequential Linear Programming . 12.3 The Gradient Projection Method. 12.4 Convergence Rate of the Gradient Projection Method. 405 412 414 420
Contents xiv 12.5 12.6 12.7 12.8 12.9 12.10 13 429 435 442 445 449 450 Penalty and Barrier Methods. 455 Penalty Methods. Barrier Methods. Lagrange Multipliers in Penalty and Barrier Methods. Newton’s Method for the Logarithmic Barrier Optimization. Newton’s Method for Equality Constrained Optimization. Conjugate Gradients and Penalty Methods. Penalty Functions and Gradient Projection. Summary. Exercises. 456 460 463 470 473 476 477 481 482 Local Duality and Dual Methods. 487 488 494 498 503 508 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14 The Reduced Gradient Method. Convergence Rate of the Reduced Gradient Method. Sequential Quadratic Optimization Methods. Active Set Methods. Summary.
Exercises. 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Local Duality and the Lagrangian Method. Separable Problems and Their Duals. The Augmented Lagrangian and Interpretation. The Augmented Lagrangian Method of Multipliers . The Alternating Direction Method of Multipliers. The Multi-Block Extension of the Alternating Direction Method of Multipliers. 513 *Cutting Plane Methods. Exercises. 515 521 15 Primal-Dual Methods. 525 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 A The Standard Problem and Monotone Function. A Simple Merit Function. Basic Primal-Dual Methods. Relation to Sequential Quadratic Optimization . Primal-Dual Interior-Point (Barrier) Methods. The Monotone Complementarity Problem. Detect Infeasibility in Nonlinear Optimization.
Summary. Exercises. 525 529 531 537 542 547 550 553 554 Mathematical Review. 559 A.l A.2 A.3 A.4 A.5 A.6 Sets . Matrix Notation. Spaces. Eigenvalues and Quadratic Forms. Topological Concepts. Functions. 559 560 561 562 564 564
Contents В Convex Sets. 571 B.l B.2 B.3 B.4 C Basic Definitions. Hyperplanes and Polytopes. Separating and Supporting Hyperplanes. Extreme Points. 571 573 575 577 Gaussian Elimination. 579 C.l C.2 D XV The LU Decomposition. 579 Pivots. 582 Basic Network Concepts. 587 Flows in Networks. Tree Procedure. Capacitated Networks . 589 589 591 Bibliography. 593 Index. 607 D.l D.2 D.3 |
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| author | Luenberger, David G. 1937- Ye, Yinyu |
| author_GND | (DE-588)135911338 (DE-588)170450880 |
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| bvnumber | BV048415764 |
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| classification_tum | WIR 000 |
| ctrlnum | (OCoLC)1346091736 (DE-599)BVBBV048415764 |
| dewey-full | 658.40301 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 658 - General management |
| dewey-raw | 658.40301 |
| dewey-search | 658.40301 |
| dewey-sort | 3658.40301 |
| dewey-tens | 650 - Management and auxiliary services |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | Fifth edition |
| format | Book |
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| id | DE-604.BV048415764 |
| illustrated | Illustrated |
| index_date | 2024-07-03T20:26:30Z |
| indexdate | 2024-07-10T09:37:36Z |
| institution | BVB |
| isbn | 9783030854492 9783030854522 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033794146 |
| oclc_num | 1346091736 |
| open_access_boolean | |
| owner | DE-384 DE-29T DE-739 |
| owner_facet | DE-384 DE-29T DE-739 |
| physical | XV, 609 Seiten Illustrationen |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | Springer |
| record_format | marc |
| series | International series in operations research & management science |
| series2 | International series in operations research & management science |
| spelling | Luenberger, David G. 1937- Verfasser (DE-588)135911338 aut Linear and Nonlinear Programming David G. Luenberger, Yinyu Ye Fifth edition Cham, Switzerland Springer [2021] XV, 609 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier International series in operations research & management science volume 228 Operations Research/Decision Theory Operations Research, Management Science Mathematical Modeling and Industrial Mathematics Engineering Economics, Organization, Logistics, Marketing Operations research Decision making Management science Mathematical models Engineering economics Engineering economy Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Lineare Ordnung (DE-588)4167706-7 gnd rswk-swf Lineare Ordnung (DE-588)4167706-7 s Nichtlineare Optimierung (DE-588)4128192-5 s DE-604 Lineare Optimierung (DE-588)4035816-1 s Optimierung (DE-588)4043664-0 s Ye, Yinyu Verfasser (DE-588)170450880 aut Erscheint auch als Online-Ausgabe 978-3-030-85450-8 Vorangegangen ist 978-3-319-18841-6 International series in operations research & management science volume 228 (DE-604)BV011630976 228 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033794146&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Luenberger, David G. 1937- Ye, Yinyu Linear and Nonlinear Programming International series in operations research & management science Operations Research/Decision Theory Operations Research, Management Science Mathematical Modeling and Industrial Mathematics Engineering Economics, Organization, Logistics, Marketing Operations research Decision making Management science Mathematical models Engineering economics Engineering economy Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd Optimierung (DE-588)4043664-0 gnd Lineare Ordnung (DE-588)4167706-7 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4128192-5 (DE-588)4043664-0 (DE-588)4167706-7 |
| title | Linear and Nonlinear Programming |
| title_auth | Linear and Nonlinear Programming |
| title_exact_search | Linear and Nonlinear Programming |
| title_exact_search_txtP | Linear and Nonlinear Programming |
| title_full | Linear and Nonlinear Programming David G. Luenberger, Yinyu Ye |
| title_fullStr | Linear and Nonlinear Programming David G. Luenberger, Yinyu Ye |
| title_full_unstemmed | Linear and Nonlinear Programming David G. Luenberger, Yinyu Ye |
| title_short | Linear and Nonlinear Programming |
| title_sort | linear and nonlinear programming |
| topic | Operations Research/Decision Theory Operations Research, Management Science Mathematical Modeling and Industrial Mathematics Engineering Economics, Organization, Logistics, Marketing Operations research Decision making Management science Mathematical models Engineering economics Engineering economy Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd Optimierung (DE-588)4043664-0 gnd Lineare Ordnung (DE-588)4167706-7 gnd |
| topic_facet | Operations Research/Decision Theory Operations Research, Management Science Mathematical Modeling and Industrial Mathematics Engineering Economics, Organization, Logistics, Marketing Operations research Decision making Management science Mathematical models Engineering economics Engineering economy Lineare Optimierung Nichtlineare Optimierung Optimierung Lineare Ordnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033794146&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011630976 |
| work_keys_str_mv | AT luenbergerdavidg linearandnonlinearprogramming AT yeyinyu linearandnonlinearprogramming |