An introduction to optimization:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley-Interscience
2008
|
| Edition: | 3. ed. |
| Series: | Wiley-Interscience series in discrete mathematics and optimization
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 563 - 570 |
| Physical Description: | XV, 584 S. graph. Darst. |
| ISBN: | 9780471758006 0471758000 |
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| 245 | 1 | 0 | |a An introduction to optimization |c Edwin K. P. Chong ; Stanislaw H. Żak |
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| adam_text |
Titel: An introduction to optimization
Autor: Chong, Edwin Kah Pin
Jahr: 2008
CONTENTS
Preface xiii
PART I MATHEMATICAL REVIEW
1 Methods of Proof and Some Notation 3
1.1 Methods of Proof 3
1.2 Notation 5
Exercises 6
2 Vector Spaces and Matrices 7
2.1 Vector and Matrix 7
2.2 Rank of a Matrix 13
2.3 Linear Equations 17
2.4 Inner Products and Norms 19
Exercises 22
3 Transformations 23
3.1 Linear Transformations 23
vii
viii contents
3.2 Eigenvalues and Eigenvectors 24
3.3 Orthogonal Projections 27
3.4 Quadratic Forms 29
3.5 Matrix Norms 33
Exercises 38
4 Concepts from Geometry 43
4.1 Line Segments 43
4.2 Hyperplanes and Linear Varieties 44
4.3 Convex Sets 46
4.4 Neighborhoods 48
4.5 Poly topes and Polyhedra 50
Exercises 51
5 Elements of Calculus 53
5.1 Sequences and Limits 53
5.2 Differentiability 60
5.3 The Derivative Matrix 61
5.4 Differentiation Rules 65
5.5 Level Sets and Gradients 66
5.6 Taylor Series 70
Exercises 75
PART II UNCONSTRAINED OPTIMIZATION
6 Basics of Set-Constrained and Unconstrained Optimization 79
6.1 Introduction 79
6.2 Conditions for Local Minimizers 81
Exercises 91
7 One-Dimensional Search Methods 101
7.1 Golden Section Search 101
7.2 Fibonacci Search 105
7.3 Newton's Method 113
7.4 Secant Method 117
7.5 Remarks on Line Search Methods 119
Exercises 121
CONTENTS IX
8 Gradient Methods 125
8.1 Introduction 125
8.2 The Method of Steepest Descent 127
8.3 Analysis of Gradient Methods 135
Exercises 147
9 Newton's Method 155
9.1 Introduction 155
9.2 Analysis of Newton's Method 158
9.3 Levenberg-Marquardt Modification 162
9.4 Newton's Method for Nonlinear Least Squares 162
Exercises 165
10 Conjugate Direction Methods 169
10.1 Introduction 169
10.2 The Conjugate Direction Algorithm 171
10.3 The Conjugate Gradient Algorithm 176
10.4 The Conjugate Gradient Algorithm for Nonquadratic
Problems 180
Exercises 182
11 Quasi-Newton Methods 187
11.1 Introduction 187
11.2 Approximating the Inverse Hessian 188
11.3 The Rank One Correction Formula 191
11.4 The DFP Algorithm 196
11.5 The BFGS Algorithm 201
Exercises 205
12 Solving Linear Equations 211
12.1 Least-Squares Analysis 211
12.2 The Recursive Least-Squares Algorithm 221
12.3 Solution to a Linear Equation with Minimum Norm 225
12.4 Kaczmarz's Algorithm 226
12.5 Solving Linear Equations in General 230
Exercises 238
X CONTENTS
13 Unconstrained Optimization and Neural Networks 247
13.1 Introduction 247
13.2 Single-Neuron Training 250
13.3 The Backpropagation Algorithm 252
Exercises 264
14 Global Search Algorithms 267
14.1 Introduction 267
14.2 The Nelder-Mead Simplex Algorithm 268
14.3 Simulated Annealing 272
14.4 Particle Swarm Optimization 276
14.5 Genetic Algorithms 279
Exercises 292
PART III LINEAR PROGRAMMING
15 Introduction to Linear Programming 299
15.1 Brief History of Linear Programming 299
15.2 Simple Examples of Linear Programs 301
15.3 Two-Dimensional Linear Programs 308
15.4 Convex Polyhedra and Linear Programming 310
15.5 Standard Form Linear Programs 312
15.6 Basic Solutions 318
15.7 Properties of Basic Solutions 321
15.8 Geometric View of Linear Programs 324
Exercises 329
16 Simplex Method 333
16.1 Solving Linear Equations Using Row Operations 333
16.2 The Canonical Augmented Matrix 340
16.3 Updating the Augmented Matrix 342
16.4 The Simplex Algorithm 343
16.5 Matrix Form of the Simplex Method 350
16.6 Two-Phase Simplex Method 354
16.7 Revised Simplex Method 358
Exercises 363
17 Duality 371
CONTENTS Xi
17.1 Dual Linear Programs 371
17.2 Properties of Dual Problems 379
Exercises 386
18 Nonsimplex Methods 395
18.1 Introduction 395
18.2 Khachiyan's Method 397
18.3 Affine Scaling Method 400
18.4 Karmarkar's Method 405
Exercises 418
PART IV NONLINEAR CONSTRAINED OPTIMIZATION
19 Problems with Equality Constraints 423
19.1 Introduction 423
19.2 Problem Formulation 425
19.3 Tangent and Normal Spaces 426
19.4 Lagrange Condition 433
19.5 Second-Order Conditions 442
19.6 Minimizing Quadratics Subject to Linear Constraints 446
Exercises 450
20 Problems with Inequality Constraints 457
20.1 Karush-Kuhn-Tucker Condition 457
20.2 Second-Order Conditions 466
Exercises 471
21 Convex Optimization Problems 479
21.1 Introduction 479
21.2 Convex Functions 482
21.3 Convex Optimization Problems 491
21.4 Semidefinite Programming 497
Exercises 506
22 Algorithms for Constrained Optimization 513
22.1 Introduction 513
22.2 Projections 513
22.3 Projected Gradient Methods with Linear Constraints 517
Xii CONTENTS
22.4 Lagrangian Algorithms 521
22.5 Penalty Methods 528
Exercises 535
23 Multiobjective Optimization 541
23.1 Introduction 541
23.2 Pareto Solutions 542
23.3 Computing the Pareto Front 545
23.4 From Multiobjective to Single-Objective Optimization 549
23.5 Uncertain Linear Programming Problems 552
Exercises 560
References 563
Index 571 |
| adam_txt |
Titel: An introduction to optimization
Autor: Chong, Edwin Kah Pin
Jahr: 2008
CONTENTS
Preface xiii
PART I MATHEMATICAL REVIEW
1 Methods of Proof and Some Notation 3
1.1 Methods of Proof 3
1.2 Notation 5
Exercises 6
2 Vector Spaces and Matrices 7
2.1 Vector and Matrix 7
2.2 Rank of a Matrix 13
2.3 Linear Equations 17
2.4 Inner Products and Norms 19
Exercises 22
3 Transformations 23
3.1 Linear Transformations 23
vii
viii contents
3.2 Eigenvalues and Eigenvectors 24
3.3 Orthogonal Projections 27
3.4 Quadratic Forms 29
3.5 Matrix Norms 33
Exercises 38
4 Concepts from Geometry 43
4.1 Line Segments 43
4.2 Hyperplanes and Linear Varieties 44
4.3 Convex Sets 46
4.4 Neighborhoods 48
4.5 Poly topes and Polyhedra 50
Exercises 51
5 Elements of Calculus 53
5.1 Sequences and Limits 53
5.2 Differentiability 60
5.3 The Derivative Matrix 61
5.4 Differentiation Rules 65
5.5 Level Sets and Gradients 66
5.6 Taylor Series 70
Exercises 75
PART II UNCONSTRAINED OPTIMIZATION
6 Basics of Set-Constrained and Unconstrained Optimization 79
6.1 Introduction 79
6.2 Conditions for Local Minimizers 81
Exercises 91
7 One-Dimensional Search Methods 101
7.1 Golden Section Search 101
7.2 Fibonacci Search 105
7.3 Newton's Method 113
7.4 Secant Method 117
7.5 Remarks on Line Search Methods 119
Exercises 121
CONTENTS IX
8 Gradient Methods 125
8.1 Introduction 125
8.2 The Method of Steepest Descent 127
8.3 Analysis of Gradient Methods 135
Exercises 147
9 Newton's Method 155
9.1 Introduction 155
9.2 Analysis of Newton's Method 158
9.3 Levenberg-Marquardt Modification 162
9.4 Newton's Method for Nonlinear Least Squares 162
Exercises 165
10 Conjugate Direction Methods 169
10.1 Introduction 169
10.2 The Conjugate Direction Algorithm 171
10.3 The Conjugate Gradient Algorithm 176
10.4 The Conjugate Gradient Algorithm for Nonquadratic
Problems 180
Exercises 182
11 Quasi-Newton Methods 187
11.1 Introduction 187
11.2 Approximating the Inverse Hessian 188
11.3 The Rank One Correction Formula 191
11.4 The DFP Algorithm 196
11.5 The BFGS Algorithm 201
Exercises 205
12 Solving Linear Equations 211
12.1 Least-Squares Analysis 211
12.2 The Recursive Least-Squares Algorithm 221
12.3 Solution to a Linear Equation with Minimum Norm 225
12.4 Kaczmarz's Algorithm 226
12.5 Solving Linear Equations in General 230
Exercises 238
X CONTENTS
13 Unconstrained Optimization and Neural Networks 247
13.1 Introduction 247
13.2 Single-Neuron Training 250
13.3 The Backpropagation Algorithm 252
Exercises 264
14 Global Search Algorithms 267
14.1 Introduction 267
14.2 The Nelder-Mead Simplex Algorithm 268
14.3 Simulated Annealing 272
14.4 Particle Swarm Optimization 276
14.5 Genetic Algorithms 279
Exercises 292
PART III LINEAR PROGRAMMING
15 Introduction to Linear Programming 299
15.1 Brief History of Linear Programming 299
15.2 Simple Examples of Linear Programs 301
15.3 Two-Dimensional Linear Programs 308
15.4 Convex Polyhedra and Linear Programming 310
15.5 Standard Form Linear Programs 312
15.6 Basic Solutions 318
15.7 Properties of Basic Solutions 321
15.8 Geometric View of Linear Programs 324
Exercises 329
16 Simplex Method 333
16.1 Solving Linear Equations Using Row Operations 333
16.2 The Canonical Augmented Matrix 340
16.3 Updating the Augmented Matrix 342
16.4 The Simplex Algorithm 343
16.5 Matrix Form of the Simplex Method 350
16.6 Two-Phase Simplex Method 354
16.7 Revised Simplex Method 358
Exercises 363
17 Duality 371
CONTENTS Xi
17.1 Dual Linear Programs 371
17.2 Properties of Dual Problems 379
Exercises 386
18 Nonsimplex Methods 395
18.1 Introduction 395
18.2 Khachiyan's Method 397
18.3 Affine Scaling Method 400
18.4 Karmarkar's Method 405
Exercises 418
PART IV NONLINEAR CONSTRAINED OPTIMIZATION
19 Problems with Equality Constraints 423
19.1 Introduction 423
19.2 Problem Formulation 425
19.3 Tangent and Normal Spaces 426
19.4 Lagrange Condition 433
19.5 Second-Order Conditions 442
19.6 Minimizing Quadratics Subject to Linear Constraints 446
Exercises 450
20 Problems with Inequality Constraints 457
20.1 Karush-Kuhn-Tucker Condition 457
20.2 Second-Order Conditions 466
Exercises 471
21 Convex Optimization Problems 479
21.1 Introduction 479
21.2 Convex Functions 482
21.3 Convex Optimization Problems 491
21.4 Semidefinite Programming 497
Exercises 506
22 Algorithms for Constrained Optimization 513
22.1 Introduction 513
22.2 Projections 513
22.3 Projected Gradient Methods with Linear Constraints 517
Xii CONTENTS
22.4 Lagrangian Algorithms 521
22.5 Penalty Methods 528
Exercises 535
23 Multiobjective Optimization 541
23.1 Introduction 541
23.2 Pareto Solutions 542
23.3 Computing the Pareto Front 545
23.4 From Multiobjective to Single-Objective Optimization 549
23.5 Uncertain Linear Programming Problems 552
Exercises 560
References 563
Index 571 |
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| index_date | 2024-07-02T20:35:37Z |
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| spelling | Chong, Edwin K. P. 1966- Verfasser (DE-588)13220973X aut An introduction to optimization Edwin K. P. Chong ; Stanislaw H. Żak 3. ed. Hoboken, NJ Wiley-Interscience 2008 XV, 584 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley-Interscience series in discrete mathematics and optimization Literaturverz. S. 563 - 570 Optimización matemática Optimierung (DE-588)4043664-0 gnd rswk-swf (DE-588)4006432-3 Bibliografie gnd-content Optimierung (DE-588)4043664-0 s DE-604 Żak, Stanislaw H. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016455188&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chong, Edwin K. P. 1966- Żak, Stanislaw H. An introduction to optimization Optimización matemática Optimierung (DE-588)4043664-0 gnd |
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| title | An introduction to optimization |
| title_auth | An introduction to optimization |
| title_exact_search | An introduction to optimization |
| title_exact_search_txtP | An introduction to optimization |
| title_full | An introduction to optimization Edwin K. P. Chong ; Stanislaw H. Żak |
| title_fullStr | An introduction to optimization Edwin K. P. Chong ; Stanislaw H. Żak |
| title_full_unstemmed | An introduction to optimization Edwin K. P. Chong ; Stanislaw H. Żak |
| title_short | An introduction to optimization |
| title_sort | an introduction to optimization |
| topic | Optimización matemática Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Optimización matemática Optimierung Bibliografie |
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