Linear complementarity, linear and nonlinear programming:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin
Heldermann Verlag Berlin
1988
|
| Schriftenreihe: | Sigma series in applied mathematics
3 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xlviii, 629 Seiten Diagramme |
| ISBN: | 3885384035 |
Internformat
MARC
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| 100 | 1 | |a Murty, Katta G. |d 1936- |0 (DE-588)1060328593 |4 aut | |
| 245 | 1 | 0 | |a Linear complementarity, linear and nonlinear programming |c K. G. Murty |
| 264 | 1 | |a Berlin |b Heldermann Verlag Berlin |c 1988 | |
| 300 | |a xlviii, 629 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Sigma series in applied mathematics |v 3 | |
| 650 | 4 | |a Algorithme | |
| 650 | 4 | |a Algèbre linéaire | |
| 650 | 4 | |a Complexité algorithme | |
| 650 | 4 | |a Convexité | |
| 650 | 4 | |a Exercice programmation | |
| 650 | 4 | |a Jeu | |
| 650 | 4 | |a Méthode itérative | |
| 650 | 4 | |a Problèmes de la complémentarité linéaire - Guides, manuels, etc | |
| 650 | 4 | |a Programmation linéaire | |
| 650 | 4 | |a Programmation linéaire - Guides, manuels, etc | |
| 650 | 7 | |a Programmation linéaire |2 ram | |
| 650 | 4 | |a Programmation non linéaire - Guides, manuels, etc | |
| 650 | 4 | |a Programmation quadratique | |
| 650 | 4 | |a Linear complementarity problem | |
| 650 | 4 | |a Linear programming | |
| 650 | 4 | |a Nonlinear programming | |
| 650 | 0 | 7 | |a Nichtlineare Optimierung |0 (DE-588)4128192-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare Optimierung |0 (DE-588)4035816-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineares Komplementaritätsproblem |0 (DE-588)4198983-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lineares Komplementaritätsproblem |0 (DE-588)4198983-1 |D s |
| 689 | 0 | 1 | |a Nichtlineare Optimierung |0 (DE-588)4128192-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Lineares Komplementaritätsproblem |0 (DE-588)4198983-1 |D s |
| 689 | 1 | 1 | |a Lineare Optimierung |0 (DE-588)4035816-1 |D s |
| 689 | 1 | |5 DE-604 | |
| 830 | 0 | |a Sigma series in applied mathematics |v 3 |w (DE-604)BV001902107 |9 3 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents ix
CONTENTS
1 LINEAR COMPLEMENTARITY PROBLEM,
ITS GEOMETRY AND APPLICATIONS 1
1.1 The linear complementarity problem and its geometry 1
1.1.1 Notation 3
1.1.2 Complementary cones 4
1.1.3 The linear complementarity problem 6
1.2 Application to linear programming 9
1.3 Quadratic programming 11
1.3.1 Review on positive semidefinite matrices 11
1.3.2 Relationship of positive semidefiniteness to the convexity
of quadratic functions 23
1.3.3 Necessary optimality conditions for quadratic
programming 24
1.3.4 Convex quadratic programs and LCP s associated with
PSD matrices 28
1.3.5 Applications of quadratic programming 29
1.3.6 Application of quadratic programming in algorithms for
NLP, recursive quadratic programming methods for NLP .. 31
1.4 Two person games 40
1.5 Other applications 44
1.6 The Nonlinear Complementarity Problem 44
1.7 Exercises 45
1.8 References 57
2 THE COMPLEMENTARY PIVOT ALGORITHM
AND ITS EXTENSION TO FIXED POINT
COMPUTING 62
2.1 Bases and basic feasible solutions 63
2.2 The complementary pivot algorithm 66
2.2.1 The original tableau 67
2.2.2 Pivot steps 67
2.2.3 Initialization 71
2.2.4 Almost complementary feasible basic vectors 71
x Contents
2.2.5 Complementary pivot rule 72
2.2.6 Termination 74
2.2.7 Implementation of the complementary pivot method
using the inverse of the basis 80
2.2.S Cycling under degenerary in the complementary
pivot method 83
2.3 Conditions under which the complementary pivot
method works 85
2.3.1 Results on LCP s associated with copositive plus
matrices 86
2.3.2 Results on LCPs associated with L and £* matrices 89
2.3.3 A variant of the complementary pivot algorithm 97
2.3.4 Lexicographic Lemke algorithm 99
2.3.5 Another sufficient condition for the complementary
pivot method to process the LCP (q,M) 99
2.3.6 Unboundedness of the objective function 101
2.3.7 Some results on complementary BFS s 104
2.4 A method for carrying out the complementary
pivot algorithm without introducing any artificial
variables, under certain conditions 105
2.5 To find an equilibrium pair of strategies for a
bimatrix game game using the complementary
pivot algorithm 108
2.6 A variable dimension algorithm 115
2.7 Extensions to fixed point computing methods,
piecewise linear and simplicial methods 123
2.7.1 Some definitions 124
2.7.2 A review of some fixed point theorems 128
2.7.3 Application in unconstrained optimization 137
2.7.4 Application to solve a system of nonlinear inequalities .... 138
2.7.5 Application to solve a system of nonlinear equations 138
2.7.6 Application to solve the nonlinear programming
problem 139
2.7.7 Application to solve the nonlinear complementarity
problem 142
2.7.8 Merrill s algorithm for computing Kakutani fixed points .. 143
2.8 Computational complexity of the complementary
pivot algorithm 160
CONTENTS Xi
2.9 The general quadratic programming problem 163
2.0.1 Testing copositiveness 165
2.0.2 Computing a KKT point for a general quadratic
programming problem 166
2.0.3 Computing a global minimum, or even a local minimum,
in nonconvex programming problems may be hard 170
2.10 Exercises 180
2.11 References 188
3 SEPARATION PROPERTIES, PRINCIPAL PIVOT
TRANSFORMS, CLASSES OF MATRICES 195
3.1 LCP s associated with principally nondegenerate
matrices 196
3.2 Principal pivot transforms 199
3.2.1 Principal rearrangements of a square matrix 208
3.3 LCP s associated with P matrices 209
3.3.1 One to one correspondence between complementary
bases and sign vectors 223
3.4 Other classes of matrices in the study of the LCP 224
3.5 Exercises 231
3.6 References 249
4 PRINCIPAL PIVOTING METHODS FOR LCP 254
4.1 Principal pivoting method I 254
4.1.1 Extension to an algorithm for the nonlinear
complementarity problem 259
4.1.2 Some methods which do not work 260
4.2 The Graves principal pivoting method 263
4.3 Dantzig Cottle principal pivoting method 273
4.4 References 278
5 THE PARAMETRIC LINEAR
COMPLEMENTARITY PROBLEM 280
5.1 Parametric convex quadratic programming 289
xii Contents
5.2 Exercises 296
5.3 References 298
6 COMPUTATIONAL COMPLEXITY OF
COMPLEMENTARY PIVOT METHODS 300
6.1 Computational complexity of the parametric
LGP algorithm 303
6.2 Geometric interpretation of a pivot step in the
complementary pivot method 304
6.3 Computational complexity of the complementary
pivot method *. 306
6.4 Computational complexity of the principal pivoting
method I 307
6.5 Exercises 311
6.6 References 313
7 NEAREST POINT PROBLEMS ON
SIMPLICIAL CONES 314
7.1 Exercises 328
7.2 References 332
8 POLYNOMIALLY BOUNDED ALGORITHMS
FOR SOME CLASSES OF LCP s 333
8.1 Chandrasekaran s algorithm for LCP s associated
with Z matrices 333
8.2 A back substitution method for the LCP s associated
with triangular P matrices 335
8.3 Polynomially bounded ellipsoid algorithms for LCP s
corresponding to convex quadratic programs 336
8.4 An ellipsoid algorithm for the nearest point problem
on simplicial cones 338
8.5 An ellipsoid algorithm for LCP s associated
with PD matrices 347
Contents xiii
8.6 An ellipsoid algorithm for LCP s associated
with PSD matrices 351
8.7 Some JV^ complete classes of LCP s 354
8.8 An ellipsoid algorithm for nonlinear programming 356
8.9 Exercises 358
8.10 References 359
9 ITERATIVE METHODS FOR LCP s 361
9.1 Introduction 361
9.2 An iterative method for LCP s associated with
PD symmetric matrices 363
9.3 Iterative methods for LCP s associated with
general symmetric matrices 366
9.3.1 Application of these methods to solve convex
quadratic programs 373
0.3.2 Application to convex quadratic programs subject
to general constraints 375
0.3.3 How to apply these iterative schemes in practice 377
9.4 Sparsity preserving SOR methods for separable
quadratic programming 378
0.4.1 Application to separable convex quadratic
programming 379
9.5 Iterative methods for general LCP s 381
9.6 Iterative methods for LCP s based on matrix
splittings 383
9.7 Exercises 386
9.8 References 387
10 SURVEY OF DESCENT BASED METHODS
FOR UNCONSTRAINED AND LINEARLY
CONSTRAINED MINIMIZATION 389
10.1 A formulation example for a linearly constrained
nonlinear program 391
10.2 Types of solutions for a nonlinear program 394
xhr Contents
10.3 What can and cannot be done efficiently by
existing methods 395
10.4 Can we at least find a local minimum ? 397
10.5 Precision in computation 398
10.6 Rates of convergence 399
10.7 Survey of some line minimization algorithms 400
10.7.1 The Golden Section Search Method 402
10.7.2 The method of bisection 403
10.7.S Newton s method 403
10.7.4 Modified Newton s method 404
10.7.5 Secant method 405
10.7.6 The method of false position 405
10.7.7 Univariate minimization by polynomial approximation
methods 406
10.7.8 Practical termination conditions for line minimization
algorithms 410
10.7.0 Line minimization algorithms based on piecewise linear
and quadratic approximations 410
10.8 Survey of descent methods for unconstrained
minimzation in R 421
10.8.1 How to determine the step length 422
10.8.2 The various methods 426
10.8.3 The method of steepest descent 426
10.8.4 Newton s method 426
10.8.5 Modified Newton s method 428
10.8.6 QuasiNewton methods 428
10.8.7 Conjugate direction methods 431
10.8.8 Practical termination conditions for unconstrained
minimization algorithms 433
10.9 Survey of some methods for linear equality constrained
minimization in R 434
10.9.1 Computing the Lagrange multiplier vector 436
10.10 Survey of optimization subject to general
linear constraints 437
10.10.1 The use of Lagrange multipliers to identify active
inequality constraints 437
iu.20.2 The general problem 438
10.10.3 The Prank Wolfe method 439
Contents xv
10.10.4 Reduced gradient methods 444
10.10.5 The gradient projection methods 445
10.10.6 The active set methods 447
10.11 Exercises 448
10.12 References 458
11 NEW LINEAR PROGRAMMING
ALGORITHMS, AND SOME OPEN PROBLEMS
IN LINEAR COMPLEMENTARITY 461
11.1 Classification of a given square matrix M 461
11.2 Worst case computational complexity of algorithms 462
11.2.1 Computational complexity of the LCP associated
with a P matrix 463
11.2.2 A principal pivoting descent algorithm for the LCP
associated with a P matrix 463
11.3 Alternate solutions of the LCP (q,M) 465
11.4 New approaches for linear programming 468
11.4.1 The Karmarkar s algorithm for linear
programming 469
11.4.2 Tardo s new strongly polynomial minimum cost
circulation algorithm 495
11.4.3 The ellipsoid method for linear programming 495
11.4.4 The gravitational method for linear programming 498
11.5 References 505
APPENDIX : PRELIMINARIES 507
1. Theorems of alternatives for systems of linear constraints 507
2. Convex sets 519
3. Convex, concave functions, their properties 531
4. Optimality conditions for smooth optimisation problems 542
5. Summary of some optimality conditions 567
6. Exercises 570
7. References 604
|
| any_adam_object | 1 |
| author | Murty, Katta G. 1936- |
| author_GND | (DE-588)1060328593 |
| author_facet | Murty, Katta G. 1936- |
| author_role | aut |
| author_sort | Murty, Katta G. 1936- |
| author_variant | k g m kg kgm |
| building | Verbundindex |
| bvnumber | BV002212752 |
| callnumber-first | T - Technology |
| callnumber-label | T57 |
| callnumber-raw | T57.74 |
| callnumber-search | T57.74 |
| callnumber-sort | T 257.74 |
| callnumber-subject | T - General Technology |
| classification_rvk | SK 870 |
| classification_tum | MAT 916f MAT 912f MAT 910f |
| ctrlnum | (OCoLC)18514726 (DE-599)BVBBV002212752 |
| dewey-full | 519.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.7 |
| dewey-search | 519.7 |
| dewey-sort | 3519.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV002212752 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:42:09Z |
| institution | BVB |
| isbn | 3885384035 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001453284 |
| oclc_num | 18514726 |
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| physical | xlviii, 629 Seiten Diagramme |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | Heldermann Verlag Berlin |
| record_format | marc |
| series | Sigma series in applied mathematics |
| series2 | Sigma series in applied mathematics |
| spelling | Murty, Katta G. 1936- (DE-588)1060328593 aut Linear complementarity, linear and nonlinear programming K. G. Murty Berlin Heldermann Verlag Berlin 1988 xlviii, 629 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Sigma series in applied mathematics 3 Algorithme Algèbre linéaire Complexité algorithme Convexité Exercice programmation Jeu Méthode itérative Problèmes de la complémentarité linéaire - Guides, manuels, etc Programmation linéaire Programmation linéaire - Guides, manuels, etc Programmation linéaire ram Programmation non linéaire - Guides, manuels, etc Programmation quadratique Linear complementarity problem Linear programming Nonlinear programming Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Lineares Komplementaritätsproblem (DE-588)4198983-1 gnd rswk-swf Lineares Komplementaritätsproblem (DE-588)4198983-1 s Nichtlineare Optimierung (DE-588)4128192-5 s DE-604 Lineare Optimierung (DE-588)4035816-1 s Sigma series in applied mathematics 3 (DE-604)BV001902107 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001453284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Murty, Katta G. 1936- Linear complementarity, linear and nonlinear programming Sigma series in applied mathematics Algorithme Algèbre linéaire Complexité algorithme Convexité Exercice programmation Jeu Méthode itérative Problèmes de la complémentarité linéaire - Guides, manuels, etc Programmation linéaire Programmation linéaire - Guides, manuels, etc Programmation linéaire ram Programmation non linéaire - Guides, manuels, etc Programmation quadratique Linear complementarity problem Linear programming Nonlinear programming Nichtlineare Optimierung (DE-588)4128192-5 gnd Lineare Optimierung (DE-588)4035816-1 gnd Lineares Komplementaritätsproblem (DE-588)4198983-1 gnd |
| subject_GND | (DE-588)4128192-5 (DE-588)4035816-1 (DE-588)4198983-1 |
| title | Linear complementarity, linear and nonlinear programming |
| title_auth | Linear complementarity, linear and nonlinear programming |
| title_exact_search | Linear complementarity, linear and nonlinear programming |
| title_full | Linear complementarity, linear and nonlinear programming K. G. Murty |
| title_fullStr | Linear complementarity, linear and nonlinear programming K. G. Murty |
| title_full_unstemmed | Linear complementarity, linear and nonlinear programming K. G. Murty |
| title_short | Linear complementarity, linear and nonlinear programming |
| title_sort | linear complementarity linear and nonlinear programming |
| topic | Algorithme Algèbre linéaire Complexité algorithme Convexité Exercice programmation Jeu Méthode itérative Problèmes de la complémentarité linéaire - Guides, manuels, etc Programmation linéaire Programmation linéaire - Guides, manuels, etc Programmation linéaire ram Programmation non linéaire - Guides, manuels, etc Programmation quadratique Linear complementarity problem Linear programming Nonlinear programming Nichtlineare Optimierung (DE-588)4128192-5 gnd Lineare Optimierung (DE-588)4035816-1 gnd Lineares Komplementaritätsproblem (DE-588)4198983-1 gnd |
| topic_facet | Algorithme Algèbre linéaire Complexité algorithme Convexité Exercice programmation Jeu Méthode itérative Problèmes de la complémentarité linéaire - Guides, manuels, etc Programmation linéaire Programmation linéaire - Guides, manuels, etc Programmation non linéaire - Guides, manuels, etc Programmation quadratique Linear complementarity problem Linear programming Nonlinear programming Nichtlineare Optimierung Lineare Optimierung Lineares Komplementaritätsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001453284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001902107 |
| work_keys_str_mv | AT murtykattag linearcomplementaritylinearandnonlinearprogramming |