Global optimization: deterministic approaches ; with 7 tables
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Edition: | 3., rev. and enl. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 681 - 718 |
| Physical Description: | XVIII, 727 S. graph. Darst. |
| ISBN: | 3540610383 |
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MARC
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| 100 | 1 | |a Horst, Reiner |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Global optimization |b deterministic approaches ; with 7 tables |c Reiner Horst ; Hoang Tuy |
| 250 | |a 3., rev. and enl. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1996 | |
| 300 | |a XVIII, 727 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 681 - 718 | ||
| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Nonlinear programming | |
| 650 | 0 | 7 | |a Globale Optimierung |0 (DE-588)4140067-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Globale Optimierung |0 (DE-588)4140067-7 |D s |
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| 700 | 1 | |a Tuy, Hoang |e Verfasser |4 aut | |
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Record in the Search Index
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| adam_text |
CONTENTS
PART A: INTRODUCTION AND BASIC TECHNIQUES 1
CHAPTER I. SOME IMPORTANT CLASSES OF GLOBAL
OPTIMIZATION PROBLEMS 3
1. Global Optimization 3
2. Concave Minimization 9
2.1. Definition and Basic Properties 9
2.2. Brief Survey of Direct Applications 12
2.3. Integer Programming and Concave Minimization 14
2.4. Bilinear Programming and Concave Minimization 20
2.5. Complementarity Problems and Concave Minimization 24
2.6. Max—Min Problems and Concave Minimization 25
3. D.C. Programming and Reverse Convex Constraints 26
3.1. D.C. Programming: Basic Properties 26
3.2. D.C. Programming: Applications 32
3.3. Reverse Convex Constraints 37
3.4. Canonical D.C. Programming Problems 39
4. Lipscbitsrian Optimization and Systems of Equations
and Inequalities 43
4.1. Lipschitzian Optimization 43
4.2. Systems of Equations and Inequalities 47
CHAPTER H. OUTER APPROXIMATION 53
1. Basic Outer Approximation Method 53
2. Outer Approximation by Convex Polyhedral Sets 58
3. Constraint Dropping Strategies 67
4. On Solving the Subproblems (Qk) 71
4.1. Finding an Initial Polytope D., and its Vertex Set V^ 72
4.2. Computing New Vertices and New Extreme Directions 74
4.3. Identifying Redundant Constraints 84
XIV
CHAPTER m. CONCAVITY CUT 89
1. Concept of a Valid Cut 89
2. Valid Cuts in the Degenerate Case 95
3. Convergence of Cutting Procedures 99
4. Concavity Cuts foi Handling Reverse Convex Constraints 104
5. A Class of Generalized Concavity Cuts 108
6. Cuts Using Negative Edge Extensions 113
CHAPTER IV. BRANCH AND BOUND 115
1. A Prototype Branch and Bound Method 115
2. Finiteness and Convergence Conditions 126
3. Typical Partition Sets and their Refinement 137
3.1. Simplices 137
3.2. Rectangles and Polyhedral Cones 142
4. Lower Bounds 145
4.1. Lipschitzian Optimization 145
4.2. Vertex Minima 146
4.3. Convex Subfunctionals 148
4.4. Duality 159
4.5. Consistency 164
5. Deletion by Infeasibility 169
6. Restart Branch and Bound Algorithm 176
PART B: CONCAVE MINIMIZATION 179
CHAPTER V. CUTTING METHODS 181
1. A Pure Cutting Algorithm 181
1.1. Valid Cuts and a Sufficient Condition
for Global Optimally 182
1.2. Outline of the Method 187
2. Facial Cut Algorithm 190
2.1. The Basic Idea 190
2.2. Finding an Extreme Face of D Relative to M 192
2.3. Facial Valid Cuts 196
2.4. A Finite Cutting Algorithm 198
XV
3. Cut and Split Algorithm 201
3.1. Partition of a Cone 202
3.2. Outline of the Method 203
3.3. Remarks 206
4. Generating Deep Cuts: The Case of Concave
Quadratic Functional 211
4.1. A Hierarchy of Valid Cuts 211
4.2. Konno's Cutting Method for Concave Quadratic Programming 217
4.3. Bilinear Programming Cuts 222
CHAPTER VI. SUCCESSIVE APPROXIMATION METHODS 225
1. Outer Approximation Algorithms 225
1.1. Linearly Constrained Problem 226
1.2. Problems with Convex Constraints 234
1.3. Reducing the Sizes of the Relaxed Problems 240
2. Inner Approximation 243
2.1. The (DG) Problem 244
2.2. The Concept of Polyhedral Annexation 245
2.3. Computing the Facets of a Polytope 248
2.4. A Polyhedral Annexation Algorithm 251
2.5. Relations to Other Methods 261
2.6. Extensions 264
3. Convex Underestimation 267
3.1. Relaxation and Successive Underestimation 267
3.2. The Falk and Hoffman Algorithm 269
3.3. Rosen's Algorithm 273
4. Concave Polyhedral Underestimation 278
4.1. Outline of the Method 278
4.2. Computation of the Concave Underestimators 281
4.3. Computation of the Nonvertical Facets 282
4.4. Polyhedral Underestimation Algorithm 285
4.5. Alternative Interpretation 287
4.6. Separable Problems 289
CHAPTER VH. SUCCESSIVE PARTITION METHODS 295
1. Conical Algorithms 295
1.1. The Normal Conical Subdivision Process 296
1.2. The Main Subroutine 298
1.3. Construction of Normal Subdivision Processes 300
1.4. The Basic NCS Process 306
1.5. The Normal Conical Algorithm 308
1.6. Remarks Concerning Implementation 312
1.7. Example 315
1.8. Alternative Variants 319
1.9. Concave Minimization with Convex Constraints 323
1.10. Unbounded Feasible Domain 328
XVI
1.11. A Class of Exhaustive Subdivision Processes 329
1.12. Exhaustive Nondegenerate Subdivision Processes 335
2. Simplicial Algorithms 342
2.1. Normal Simplicial Subdivision Processes 343
2.2. Normal Simplicial Algorithm 345
2.3. Construction of an NSS Process 347
2.4. The Basic NSS Process 349
2.5. Normal Simplicial Algorithm for Problems with
Convex Constraints 351
3. An Exact Simplicial Algorithm 353
3.1. Simplicial Subdivision of a Polytope 353
3.2. A Finite Branch and Bound Procedure 356
3.3. A Modified ES Algorithm 358
3.4. Unbounded Feasible Set 361
4. Rectangular Algorithms 365
4.1. Normal Rectangular Algorithm 366
4.2. Construction of an NRS Process 369
4.3. Specialization to Concave Quadratic Programming 371
4.4. Example 377
CHAPTER VIH. DECOMPOSITION OF LARGE SCALE PROBLEMS 381
1. Decomposition Framework 382
2. Branch and Bound Approach 384
2.1. Normal Simplicial Algorithm 385
2.2. Normal Rectangular Algorithm 388
2.3. Normal Conical Algorithm 390
3. Polyhedral Underestimation Method 391
3.1. Nonseparable Problems 391
3.2. Separable Problems 393
4. Decomposition by Outer Approximation 401
4.1. Basic Idea 401
4.2. Decomposition Algorithm 403
4.3. An Extension 408
4.4. Outer Approximation Versus Successive Partition 412
4.5. Outer Approximation Combined with Branch and Bound 417
5. Decomposition, of Concave Minimization Problems
over Networks 421
5.1. The Minimum Concave Cost Flow Problem 421
5.2. The Single Source Uncapacitated Minimum Concave
Cost Flow (SUCF) Problem 426
5.3. Decomposition Method for (SUCF) 433
5.4. Extension 443
XVII
CHAPTER IX. SPECIAL PROBLEMS OF CONCAVE MINIMIZATION 447
1. Bilinear Programming 447
1.1. Basic Properties 448
1.2. Cutting Plane Method 451
1.3. Polyhedral Annexation 456
1.4. Conical Algorithm 458
1.5. Outer Approximation Method 462
2. Complementarity Problems 469
2.1. Basic Properties 470
2.2. Polyhedral Annexation Method for the Linear
Complementarity Problem (LCP) 472
2.3. Conical Algorithm for the (LCP) 475
2.4. Other Global Optimization Approaches to (LCP) 483.
2.5. The Concave Complementarity Problem 486
3. Parametric Concave Programming 490
3.1. Basic Properties 492
3.2. Outer Approximation Method for (LRCP) 497
3.3. Methods Based on the Edge Property 500
3.4. Conical Algorithms for (LRCP) 508
PART C: GENERAL NONLINEAR PROBLEMS 517
CHAPTER X. D.C. PROGRAMMING 519
1. Outer Approximation Methods for Solving the Canonical
D.C. Programming Problem 519
1.1. Duality between the Objective and the Constraints 520
1.2. Outer Approximation Algorithms for Canonical D.C. Problems 526
1.3. Outer Approximation for Solving Noncanonical
D.C. Problems 541
2. Branch and Bound Methods 553
3. Solving D.C. Problems by a Sequence of Linear Programs
and Line Searches 558
4. Some Special D.C. Problems and Applications 572
4.1. The Design Centering Problem 572
4.2. The Diamond Cutting Problem 581
4.3. Biconvex Programming and Related Problems 592
CHAPTER XL LIPSCHITZ AND CONTINUOUS OPTIMIZATION 603
1. Brief Introduction into the Global Minimization
of Univariate Lipschiti Functions 604
1.1. Saw Tooth Covers 604
1.2. Algorithms for Solving the Univariate Lipschitz Problem 609
XVIII
2. Branch and Bound Algorithms 616
2.1. Branch and Bound Interpretation of Piyavskii's
Univariate Algorithm 617
2.2. Branch and Bound Methods for Minimizing a Lipschitz Function
over an n—dimensional Rectangle 621
2.3. Branch and Bound Methods for Solving Lipschitz
Optimization Problems with General Constraints 632
2.4. Global Optimization of Concave Functions Subject
to Separable Quadratic Constraints 634
2.5. Linearly Constrained Global Optimization of
Functions with Concave Minorants 645
3. Outer Approximation 653
4. The Relief Indicator Method 662
4.1. Separators for f on D 663
4.2. A Global Optimality Criterion 666
4.3. The Relief Indicator Method 670
References 681
Notation 719
Index 723 |
| any_adam_object | 1 |
| author | Horst, Reiner Tuy, Hoang |
| author_facet | Horst, Reiner Tuy, Hoang |
| author_role | aut aut |
| author_sort | Horst, Reiner |
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| callnumber-label | QA402 |
| callnumber-raw | QA402.5.H67 1996 |
| callnumber-search | QA402.5.H67 1996 |
| callnumber-sort | QA 3402.5 H67 41996 |
| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)614207526 (DE-599)BVBBV010726639 |
| dewey-full | 519.7/620 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.7/6 20 |
| dewey-search | 519.7/6 20 |
| dewey-sort | 3519.7 16 220 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 3., rev. and enl. ed. |
| format | Book |
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| id | DE-604.BV010726639 |
| illustrated | Illustrated |
| indexdate | 2025-02-24T09:01:08Z |
| institution | BVB |
| isbn | 3540610383 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007163321 |
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| physical | XVIII, 727 S. graph. Darst. |
| publishDate | 1996 |
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| publisher | Springer |
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| spelling | Horst, Reiner Verfasser aut Global optimization deterministic approaches ; with 7 tables Reiner Horst ; Hoang Tuy 3., rev. and enl. ed. Berlin [u.a.] Springer 1996 XVIII, 727 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 681 - 718 Mathematical optimization Nonlinear programming Globale Optimierung (DE-588)4140067-7 gnd rswk-swf Globale Optimierung (DE-588)4140067-7 s DE-604 Tuy, Hoang Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007163321&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Horst, Reiner Tuy, Hoang Global optimization deterministic approaches ; with 7 tables Mathematical optimization Nonlinear programming Globale Optimierung (DE-588)4140067-7 gnd |
| subject_GND | (DE-588)4140067-7 |
| title | Global optimization deterministic approaches ; with 7 tables |
| title_auth | Global optimization deterministic approaches ; with 7 tables |
| title_exact_search | Global optimization deterministic approaches ; with 7 tables |
| title_full | Global optimization deterministic approaches ; with 7 tables Reiner Horst ; Hoang Tuy |
| title_fullStr | Global optimization deterministic approaches ; with 7 tables Reiner Horst ; Hoang Tuy |
| title_full_unstemmed | Global optimization deterministic approaches ; with 7 tables Reiner Horst ; Hoang Tuy |
| title_short | Global optimization |
| title_sort | global optimization deterministic approaches with 7 tables |
| title_sub | deterministic approaches ; with 7 tables |
| topic | Mathematical optimization Nonlinear programming Globale Optimierung (DE-588)4140067-7 gnd |
| topic_facet | Mathematical optimization Nonlinear programming Globale Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007163321&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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