Lectures on modern convex optimization: analysis, algorithms, and engineering applications
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
SIAM [u.a.]
2001
|
| Series: | MPS/SIAM series on optimization
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XVI, 488 S. graph. Darst. |
| ISBN: | 0898714915 9780898714913 |
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MARC
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| 245 | 1 | 0 | |a Lectures on modern convex optimization |b analysis, algorithms, and engineering applications |c Aharon Ben-Tal ; Arkadi Nemirovski |
| 264 | 1 | |a Philadelphia |b SIAM [u.a.] |c 2001 | |
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Record in the Search Index
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Contents
Preface xi
1 Linear Programming 1
1.1 Linear programming: Basic notions 1
1.2 Example: Tschebyshev approximation and its applications 2
1.2.1 Best uniform approximation 2
1.2.2 Application: Synthesis of filters 3
1.2.3 Filter synthesis revisited 5
1.2.4 Synthesis of arrays of antennae 7
1.3 Duality in linear programming 11
1.3.1 Certificates for solvability and insolvability 12
1.3.2 Dual to a linear programming program: Origin 16
1.3.3 Linear programming duality theorem 17
1.3.4 Illustration: Problem dual to the Tschebyshev
approximation problem 19
1.3.5 Application: Truss topology design 21
1.4 Exercises to Lecture 1 30
1.4.1 Uniform approximation 30
1.4.2 Theorem on the alternative 34
1.4.3 Proof of the homogeneous Farkas lemma 35
1.4.4 Helley theorem 38
1.4.5 How many bars are needed in an optimal truss? 40
2 From Linear Programming to Conic Programming 43
2.1 Orderings of Rm and convex cones 44
2.2 What is conic programming? 46
2.3 Conic duality 50
2.3.1 Geometry of the primal and dual problems 53
2.4 Conic duality theorem 57
2.4.1 Is something wrong with conic duality? 60
2.4.2 Consequences of the conic duality theorem 61
2.4.3 Robust solvability status 64
2.5 Conic duality revisited 67
2.6 Exercises to Lecture 2 73
vii
viii Contents
2.6.1 Cones 73
2.6.2 Conic problems 76
2.6.3 Feasible and level sets of conic problems 77
3 Conic Quadratic Programming 79
3.1 Conic quadratic problems: Preliminaries 79
3.2 Examples of conic quadratic problems 81
3.2.1 Best linear approximation of complex valued functions. . 81
3.2.2 Contact problems with static friction 82
3.3 What can be expressed via conic quadratic constraints? 85
3.3.1 More examples of conic quadratic representable
functions and sets 104
3.4 More applications 109
3.4.1 Tschebyshev approximation in relative scale 109
3.4.2 Robust linear programming 110
3.4.3 Trass topology design 120
3.5 Exercises to Lecture 3 131
3.5.1 Optimal control in discrete time linear dynamic system. . 131
3.5.2 Conic quadratic representations 132
3.5.3 Does conic quadratic programming exist? 137
4 Semidefinite Programming 139
4.1 Semidefinite cone and semidefinite programs 139
4.1.1 Preliminaries 139
4.2 What can be expressed via linear matrix inequalities? 144
4.3 Applications I: Combinatorics 159
4.3.1 Shor's semidefinite relaxation scheme 161
4.3.2 Stability number, Shannon capacity, and Lovasz
capacity of a graph 164
4.3.3 MAXCUT problem 170
4.3.4 Extensions 172
4.3.5 S lemma 175
4.4 Applications II: Stability analysis 178
4.4.1 Dynamic stability in mechanics 178
4.4.2 Lyapunov stability analysis and synthesis 180
4.4.3 Interval stability analysis and synthesis 189
4.5 Applications III: Robust quadratic programming 202
4.6 Applications IV: Synthesis of filters and antennae arrays 210
4.7 Applications V: Design of chips 219
4.7.1 Building the model 220
4.7.2 Wire sizing 226
4.8 Applications VI: Structural design 227
4.8.1 Building a model 228
4.8.2 Standard case 233
4.8.3 Semidefinite reformulation of the standard SSD problem 236
4.8.4 From primal to dual 243
Contents ix
4.8.5 Back to primal 248
4.8.6 Explicit forms of the standard truss and shape problems. . 252
4.9 Applications VII: Extremal ellipsoids 257
4.9.1 Ellipsoidal approximations of unions and intersections
of ellipsoids 262
4.9.2 Approximating sums of ellipsoids 264
4.10 Exercises to Lecture 4 276
4.10.1 Around positive semidefiniteness, eigenvalues, and
: ordering 276
4.10.2 Semidefinite representations of epigraphs of convex
polynomials 291
4.10.3 Lovasz capacity number and semidefinite relaxations of
combinatorial problems 293
4.10.4 Lyapunov stability analysis 299
4.10.5 S lemma 300
4.10.6 Antenna synthesis 323
4.10.7 Ellipsoidal approximations 326
5 Computational Tractability of Convex Programs 335
5.1 Numerical solution of optimization programs—preliminaries 335
5.1.1 Mathematical programming programs 335
5.1.2 Solution methods and efficiency 336
5.2 Black box represented convex programs 342
5.3 Polynomial solvability of convex programming 352
5.3.1 Polynomial solvability of convex programming 359
5.4 Difficult problems and NP completeness 363
5.4.1 CCT—a quick introduction 363
5.4.2 From the real arithmetic complexity theory to the
CCT and back 367
6 Interior Point Polynomial Time Methods for Linear Programming,
Conic Quadratic Programming, and Semidefinite Programming 377
6.1 Motivation 377
6.1.1 Interior point methods 378
6.2 Newton method and the interior penalty scheme 379
6.2.1 Unconstrained minimization and the Newton method . . . 379
6.2.2 Classical interior penalty scheme: Construction 380
6.2.3 Classical interior penalty scheme: Drawbacks 382
6.2.4 But 382
6.3 Interior point methods for linear programming, conic quadratic
programming, and semidefinite programming: Building blocks . 384
6.3.1 Canonical cones and canonical barriers 384
6.3.2 Elementary properties of canonical barriers 387
6.4 Primal dual pair of problems and the primal dual central path 389
6.4.1 Problem(s) 389
6.4.2 Central path(s) 390
x Contents
6.5 Tracing the central path 397
6.5.1 Path following scheme 397
6.5.2 Speed of path tracing 398
6.5.3 Primal and dual path following methods 402
6.5.4 Semideflnite programming case 405
6.6 Complexity bounds for linear programming, conic quadratic
programming, and semidefinite programming 421
6.6.1 Complexity of linear programming 422
6.6.2 Complexity of conic quadratic programming 423
6.6.3 Semideflnite programming 423
6.7 Concluding remarks 424
6.8 Exercises to Lecture 6 426
6.8.1 Canonical barriers 426
6.8.2 Scalings of canonical cones 427
6.8.3 Dikin ellipsoid 429
6.8.4 More on canonical barriers 431
6.8.5 Primal path following method 432
6.8.6 Infeasible start path following method 435
Solutions to Selected Exercises 443
Exercises to Lecture 1 443
Exercises to Lecture 2 446
Exercises to Lecture 3 449
Exercises to Lecture 4 459
Exercises to Lecture 6 475
Index 485 |
| any_adam_object | 1 |
| author | Ben-Tal, Aharon Nemirovskij, Arkadij S. 1947- |
| author_GND | (DE-588)170328007 (DE-588)12209395X |
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| bvnumber | BV014091122 |
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| ctrlnum | (OCoLC)248143908 (DE-599)BVBBV014091122 |
| dewey-full | 519.7/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.7/6 |
| dewey-search | 519.7/6 |
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| discipline | Mathematik |
| format | Book |
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| indexdate | 2025-02-24T09:01:18Z |
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| isbn | 0898714915 9780898714913 |
| language | English |
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| physical | XVI, 488 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
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| publisher | SIAM [u.a.] |
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| series2 | MPS/SIAM series on optimization |
| spelling | Ben-Tal, Aharon Verfasser (DE-588)170328007 aut Lectures on modern convex optimization analysis, algorithms, and engineering applications Aharon Ben-Tal ; Arkadi Nemirovski Philadelphia SIAM [u.a.] 2001 XVI, 488 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier MPS/SIAM series on optimization Hier auch später erschienene, unveränderte Nachdrucke Konvexe Optimierung Konvexe Optimierung (DE-588)4137027-2 gnd rswk-swf Konvexe Optimierung (DE-588)4137027-2 s DE-604 Nemirovskij, Arkadij S. 1947- Verfasser (DE-588)12209395X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009652906&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ben-Tal, Aharon Nemirovskij, Arkadij S. 1947- Lectures on modern convex optimization analysis, algorithms, and engineering applications Konvexe Optimierung Konvexe Optimierung (DE-588)4137027-2 gnd |
| subject_GND | (DE-588)4137027-2 |
| title | Lectures on modern convex optimization analysis, algorithms, and engineering applications |
| title_auth | Lectures on modern convex optimization analysis, algorithms, and engineering applications |
| title_exact_search | Lectures on modern convex optimization analysis, algorithms, and engineering applications |
| title_full | Lectures on modern convex optimization analysis, algorithms, and engineering applications Aharon Ben-Tal ; Arkadi Nemirovski |
| title_fullStr | Lectures on modern convex optimization analysis, algorithms, and engineering applications Aharon Ben-Tal ; Arkadi Nemirovski |
| title_full_unstemmed | Lectures on modern convex optimization analysis, algorithms, and engineering applications Aharon Ben-Tal ; Arkadi Nemirovski |
| title_short | Lectures on modern convex optimization |
| title_sort | lectures on modern convex optimization analysis algorithms and engineering applications |
| title_sub | analysis, algorithms, and engineering applications |
| topic | Konvexe Optimierung Konvexe Optimierung (DE-588)4137027-2 gnd |
| topic_facet | Konvexe Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009652906&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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