Global optimization using interval analysis:
Mathematics of Computing -- Numerical Analysis.
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Basel ; Hong Kong ; Inc
Dekker
[1992]
|
| Schriftenreihe: | Pure and applied mathematics
165 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Mathematics of Computing -- Numerical Analysis. |
| Beschreibung: | xiii, 230 Seiten Illustrationen |
| ISBN: | 0824786963 |
Internformat
MARC
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| 100 | 1 | |a Hansen, Eldon R. |d 1927- |0 (DE-588)172125030 |4 aut | |
| 245 | 1 | 0 | |a Global optimization using interval analysis |c Eldon Hansen: Consultant, Los Altos, California |
| 264 | 1 | |a New York ; Basel ; Hong Kong ; Inc |b Dekker |c [1992] | |
| 264 | 4 | |c © 1992 | |
| 300 | |a xiii, 230 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 520 | 3 | |a Mathematics of Computing -- Numerical Analysis. | |
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Datensatz im Suchindex
| _version_ | 1804119786580344832 |
|---|---|
| adam_text | Contents
Preface v
1. INTRODUCTION 1
1.1 An Overview 1
1.2 The Origin of Interval Analysis 2
1.3 The Scope of this Book 2
1.4 Virtues and Drawbacks of Interval Mathematics 2
2. INTERVAL NUMBERS AND INTERVAL ARITHMETIC 7
2.1 Interval Numbers 7
2.2 Notation 8
2.3 Interval Arithmetic 8
2.4 Extended Interval Arithmetic 10
2.5 The Dependency Problem 11
2.6 Generalized Interval Arithmetic 12
3. FUNCTIONS OF INTERVALS 13
3.1 Real Valued Functions of Intervals 13
ix
x Contents
3.2 Interval Functions 14
3.3 The Forms of Interval Functions 16
3.4 Monotonic Functions 18
3.5 Practical Evaluation of Interval Functions 19
3.6 Thick and Thin Functions 21
4. INTERVAL LINEAR EQUATIONS 23
4.1 Definitions 23
4.2 Introduction 24
4.3 The Solution Set 25
4.4 Gaussian Elimination 26
4.5 Failure of Gaussian Elimination 29
4.6 A Preconditioning Matrix 29
4.7 The Gauss Seidel Method 31
4.8 A Special Inversion Procedure 33
5. INTERVAL LINEAR INEQUALITIES 35
5.1 Introduction 35
5.2 The Problem 36
5.3 Sets of Inequalities 38
5.4 Ordering Inequalities 40
5.5 Secondary Pivots 42
5.6 Column Interchanges 43
5.7 A Preconditioning Matrix 44
5.8 Solving Inequalities 45
6. TAYLOR SERIES AND SIMILAR EXPANSIONS 49
6.1 Introduction 49
6.2 Bounding the Remainder in Taylor Series 50
6.3 The Multidimensional Case 51
6.4 The Jacobian and Hessian 53
6.5 Automatic Differentiation 55
6.6 Sharper Bounds Using Taylor Expansions 57
6.7 The Slope Function and its Automation 58
6.8 Irrational Functions 61
6.9 Multidimensional Functions 63
7. NONLINEAR EQUATIONS OF ONE VARIABLE 67
7.1 Introduction 67
7.2 The Interval Newton Method 68
7.3 Stopping Criteria 70
7.4 The Steps of the Algorithm 74
Contents xi
7.5 Properties of the Algorithm 75
7.6 A Numerical Example 80
7.7 The Slope Interval Newton Method 81
7.8 An Example Using the Slope Method 82
7.9 Perturbed Problems 83
8. SYSTEMS OF NONLINEAR EQUATIONS 87
8.1 Introduction 87
8.2 Derivation of Interval Newton Methods 88
8.3 Variations of the Method 90
8.4 An Inner Iteration 92
8.5 Stopping Criteria 93
8.6 The Termination Process 97
8.7 Rate of Progress 98
8.8 Splitting the Box 100
8.9 The Algorithmic Steps 101
8.10 Explanation of the Algorithmic Steps 103
8.11 Properties of Interval Newton Methods 105
8.12 Perturbed Problems and Sensitivity Analysis 108
8.13 A Numerical Example 110
8.14 A Special Interval Newton Algorithm 111
9. UNCONSTRAINED OPTIMIZATION 113
9.1 Introduction 113
9.2 An Overview 115
9.3 The Initial Box 116
9.4 Use of the Gradient 116
9.5 An Upper Bound 117
9.6 A Real Search 124
9.7 A Nonconvexity Check 125
9.8 A Special Newton Process 126
9.9 Bounding the Minimum 127
9.10 Termination 128
9.11 The List of Boxes 129
9.12 Choosing a Box to Process 129
9.13 Subdividing a Box 130
9.14 The Steps of the Algorithm 132
9.15 Computational Tests 135
9.16 Conclusions 145
9.17 Nondifferentiable Problems 148
xii Preface
10. CONSTRAINED OPTIMIZATION 151
10.1 Introduction 151
10.2 The Fritz John Conditions 152
10.3 Normalization of the Lagrange Multipliers 154
10.4 Use of Constraints 156
10.5 Solving the John Conditions 157
10.6 Bounding the Lagrange Multipliers 159
10.7 First Numerical Example 161
10.8 Second Numerical Example 164
11. INEQUALITY CONSTRAINED OPTIMIZATION 167
11.1 Introduction 167
11.2 The Fritz John Conditions 169
11.3 An Upper Bound 169
11.4 A Line Search 170
11.5 Certainly Strict Feasibility 171
11.6 Linearizing the Constraints 172
11.7 The Steps of the Algorithm 174
11.8 A Numerical Example 177
11.9 Nondifferentiable Functions 178
12. EQUALITY CONSTRAINED OPTIMIZATION 179
12.1 Introduction 179
12.2 The Fritz John Conditions 179
12.3 An Upper Bound 180
12.4 Choice of Variables 181
12.5 Satisfying the Hypothesis 183
12.6 A Numerical Example 184
12.7 Linearizing the Constraints 185
12.8 The Steps of the Algorithm 186
12.9 Nondifferentiable Functions 190
13. PERTURBED PROBLEMS AND SENSITIVITY ANALYSIS 191
13.1 Introduction 191
13.2 The Basic Algorithms 193
13.3 Tolerances 194
13.4 Disjoint Solution Sets 194
13.5 Sharp Bounds for Perturbed Optimization Problems 195
13.6 Validating Assumption (A) 198
13.7 First Numerical Example 201
13.8 Second Numerical Example 203
13.9 Third Numerical Example 205
Preface xiii
13.10 An Upper Bound 207
13.11 Sharp Bounds for Perturbed Systems of Nonlinear Equations 208
14. MISCELLANY 211
14.1 Nondifferentiable Functions 211
14.2 Integer and Mixed Integer Problems 213
References 215
Author Index 223
Subject Index 227
|
| any_adam_object | 1 |
| author | Hansen, Eldon R. 1927- |
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| author_sort | Hansen, Eldon R. 1927- |
| author_variant | e r h er erh |
| building | Verbundindex |
| bvnumber | BV005574672 |
| classification_rvk | SK 870 |
| ctrlnum | (OCoLC)246624514 (DE-599)BVBBV005574672 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV005574672 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:31:44Z |
| institution | BVB |
| isbn | 0824786963 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003491496 |
| oclc_num | 246624514 |
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| physical | xiii, 230 Seiten Illustrationen |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Dekker |
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| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Hansen, Eldon R. 1927- (DE-588)172125030 aut Global optimization using interval analysis Eldon Hansen: Consultant, Los Altos, California New York ; Basel ; Hong Kong ; Inc Dekker [1992] © 1992 xiii, 230 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 165 Mathematics of Computing -- Numerical Analysis. Optimierung (DE-588)4043664-0 gnd rswk-swf Intervallanalyse (DE-588)4162150-5 gnd rswk-swf Globale Optimierung (DE-588)4140067-7 gnd rswk-swf Intervallalgebra (DE-588)4139152-4 gnd rswk-swf Globale Analysis (DE-588)4021285-3 gnd rswk-swf Globale Optimierung (DE-588)4140067-7 s Intervallanalyse (DE-588)4162150-5 s DE-604 Optimierung (DE-588)4043664-0 s Intervallalgebra (DE-588)4139152-4 s Globale Analysis (DE-588)4021285-3 s 1\p DE-604 Pure and applied mathematics 165 (DE-604)BV000001885 165 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003491496&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hansen, Eldon R. 1927- Global optimization using interval analysis Pure and applied mathematics Optimierung (DE-588)4043664-0 gnd Intervallanalyse (DE-588)4162150-5 gnd Globale Optimierung (DE-588)4140067-7 gnd Intervallalgebra (DE-588)4139152-4 gnd Globale Analysis (DE-588)4021285-3 gnd |
| subject_GND | (DE-588)4043664-0 (DE-588)4162150-5 (DE-588)4140067-7 (DE-588)4139152-4 (DE-588)4021285-3 |
| title | Global optimization using interval analysis |
| title_auth | Global optimization using interval analysis |
| title_exact_search | Global optimization using interval analysis |
| title_full | Global optimization using interval analysis Eldon Hansen: Consultant, Los Altos, California |
| title_fullStr | Global optimization using interval analysis Eldon Hansen: Consultant, Los Altos, California |
| title_full_unstemmed | Global optimization using interval analysis Eldon Hansen: Consultant, Los Altos, California |
| title_short | Global optimization using interval analysis |
| title_sort | global optimization using interval analysis |
| topic | Optimierung (DE-588)4043664-0 gnd Intervallanalyse (DE-588)4162150-5 gnd Globale Optimierung (DE-588)4140067-7 gnd Intervallalgebra (DE-588)4139152-4 gnd Globale Analysis (DE-588)4021285-3 gnd |
| topic_facet | Optimierung Intervallanalyse Globale Optimierung Intervallalgebra Globale Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003491496&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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