Primal dual interior point methods:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia, Pa.
SIAM
1997
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 289 S. graph. Darst. |
| ISBN: | 9780898713824 089871382X |
Staff View
MARC
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| 246 | 1 | 3 | |a Primal-dual interior-point methods |
| 264 | 1 | |a Philadelphia, Pa. |b SIAM |c 1997 | |
| 300 | |a XX, 289 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804125713101488128 |
|---|---|
| adam_text | Contents
Preface xiii
Notation xvii
1 Introduction 1
Linear Programming 2
Primal Dual Methods 4
The Central Path 7
A Primal Dual Framework 8
Path Following Methods 9
Potential Reduction Methods 10
Infeasible Starting Points 11
Superlinear Convergence 12
Extensions 13
Mehrotra s Predictor Corrector Algorithm 14
Linear Algebra Issues 16
Karmarkar s Algorithm 17
2 Background: Linear Programming and Interior Point
Methods 21
Standard Form 22
Optimality Conditions, Duality, and Solution Sets 23
More on Duality 24
The BliM Partition and Strict Complementarity 27
A Strictly Interior Point 29
Rank of the Matrix A 31
Bases and Vertices 32
vii
viii
Farkas s Lemma and a Proof of the Goldman Tucker Result .... 34
The Central Path 36
Background: Primal Methods 40
Primal Dual Methods: Development of the Fundamental Ideas . . 43
Notes and References 45
3 Complexity Theory 49
Polynomial Versus Exponential, Worst Case Versus Average Case . 50
Storing the Problem Data: Dimension and Size 51
The Turing Machine and Rational Arithmetic 52
Primal Dual Methods and Rational Arithmetic 52
Linear Programming and Rational Numbers 54
Moving to a Solution from an Interior Point 55
Complexity of Simplex, Ellipsoid, and Interior Point Methods ... 57
Polynomial and Strongly Polynomial Algorithms 58
Beyond the Turing Machine Model 59
More on the Real Number Model and Algebraic Complexity .... 60
A General Complexity Theorem for Path Following Methods ... 61
Notes and References 62
4 Potential Reduction Methods 65
A Primal Dual Potential Reduction Algorithm 67
Reducing $p Forces Convergence 68
A Quadratic Estimate of j p Along a Feasible Direction 70
Bounding the Coefficients in the Quadratic Approximation .... 72
An Estimate of the Reduction in $p and Polynomial Complexity . 75
What About Centrality? 78
Choosing p and a 79
Notes and References 80
5 Path Following Algorithms 83
The Short Step Path Following Algorithm 86
Technical Results 88
The Predictor Corrector Method 91
A Long Step Path Following Algorithm 96
Limit Points of the Iteration Sequence 100
Proof of Lemma 5.3 103
Notes and References 104
ix
6 Infeasible Interior Point Algorithms 107
The Algorithm 109
Convergence of Algorithm IPF Ill
Technical Results I: Bounds on i/fc||(xfc,sfc)|| 113
Technical Results II: Bounds on (Dk) 1Axk and DkAsk 115
Technical Results III: A Uniform Lower Bound on q^ 118
Proofs of Theorems 6.1 and 6.2 120
Limit Points of the Iteration Sequence 121
7 Super linear Convergence and Finite Termination 127
Affine Scaling Steps 128
An Estimate of (Ax, As): The Feasible Case 129
An Estimate of (Ax, As): The Infeasible Case 133
Algorithm PC Is Superlinear 135
Nearly Quadratic Methods 136
Convergence of Algorithm LPF f 139
Convergence of the Iteration Sequence 144
e(A, b, c) and Finite Termination 145
A Finite Termination Strategy 146
Recovering an Optimal Basis 150
More on e(A, b,c) 152
Notes and References 154
8 Extensions 157
The Monotone LCP 157
Mixed and Horizontal LCP 160
Strict Complementarity and LCP 162
Convex QP 163
Convex Programming 164
Monotone Nonlinear Complementarity and Variational Inequalities 167
Semidefinite Programming 168
Proof of Theorem 8.4 171
Notes and References 173
9 Detecting Infeasibility 177
Self Duality 178
The Simplified HSD Form 179
The HSD Form 183
Identifying a Solution Free Region 185
X
Implementations of the HSD Formulations 188
Notes and References 190
10 Practiced Aspects of Primal Dual Algorithms 193
Motivation for Mehrotra s Algorithm 194
The Algorithm 195
Superquadratic Convergence 198
Second Order Trajectory Following Methods 199
Higher Order Methods 202
Further Enhancements 204
Notes and References 207
11 Implementations 209
Three Forms of the Step Equation 210
The Cholesky Factorization 211
Sparse Cholesky Factorization: Minimum Degree Orderings .... 212
Other Orderings 215
Small Pivots in the Cholesky Factorization 216
Dense Columns in A 219
The Augmented System Formulation 220
Factoring Symmetric Indefinite Matrices 221
Starting Points 224
Termination 225
Alternative Formulations for the Linear Program 226
Free Variables 228
Presolving 230
Primal Dual Codes 232
Notes and References 232
A Basic Concepts and Results 239
Order Notation 239
Convex Sets and Functions 240
KKT Conditions 240
Convexity and Global Solutions 242
Self Duality and a Proof of Theorem 9.1 243
The Natural log Function and a Proof of Lemma 4.1 245
Singular Value Decomposition, Matrix Rank, and QR Factorization 246
Hoffman s Lemma 248
The Stewart Todd Result and a Proof of Lemma 7.2 249
xi
The Sherman Morrison Woodbury Formula 250
Asymptotic Convergence 251
Storing Sparse Matrices 253
The Turing Machine 254
B Software Packages 257
BPMPD 258
CPLEX/Barrier 259
HOPDM 259
LIPSOL 260
LOQO 261
Newton Barrier XPRESS MP 261
OSL/EKKBSLV 262
PCx 263
Bibliography 265
Index 281
|
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| dewey-sort | 3519.7 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:05:56Z |
| institution | BVB |
| isbn | 9780898713824 089871382X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007523696 |
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| physical | XX, 289 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | SIAM |
| record_format | marc |
| spelling | Wright, Stephen J. 1960- Verfasser (DE-588)121513300 aut Primal dual interior point methods Stephen J. Wright Primal-dual interior-point methods Philadelphia, Pa. SIAM 1997 XX, 289 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Innere-Punkte-Methode (DE-588)4352322-5 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 s DE-604 Innere-Punkte-Methode (DE-588)4352322-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007523696&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wright, Stephen J. 1960- Primal dual interior point methods Lineare Optimierung (DE-588)4035816-1 gnd Innere-Punkte-Methode (DE-588)4352322-5 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4352322-5 |
| title | Primal dual interior point methods |
| title_alt | Primal-dual interior-point methods |
| title_auth | Primal dual interior point methods |
| title_exact_search | Primal dual interior point methods |
| title_full | Primal dual interior point methods Stephen J. Wright |
| title_fullStr | Primal dual interior point methods Stephen J. Wright |
| title_full_unstemmed | Primal dual interior point methods Stephen J. Wright |
| title_short | Primal dual interior point methods |
| title_sort | primal dual interior point methods |
| topic | Lineare Optimierung (DE-588)4035816-1 gnd Innere-Punkte-Methode (DE-588)4352322-5 gnd |
| topic_facet | Lineare Optimierung Innere-Punkte-Methode |
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