Numerical optimization with applications:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Alpha Science Internat.
2009
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 694 S. graph. Darst. |
| ISBN: | 9781842654279 |
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| adam_text | Titel: Numerical optimization with applications
Autor: Chandra, Suresh
Jahr: 2009
Contents
Preface vii
1 Introduction 1
1.1 An Overview of Optimization 1
1.2 Optimization Problems 2
1.3 Some Simple Illustrative Examples 5
2 Linear Programming 11
2.1 Introduction 11
2.2 The Graphical Method and Some Intuitive Results 11
2.3 The Simplex Method as an Algebraic Version of The Graphical Method 16
2.4 The Simplex Method: Certain Notations 24
2.5 Description of the Simplex Method 26
2.6 Methods of Artificial Variables 33
2.7 Alternate Optima 44
2.8 Redundancy in Linear Programming 46
2.9 Degeneracy and Cycling 51
2.10 The Simplex Tableau in the Condensed Form 53
2.11 Summary and Additional Notes 55
2.12 Exercises 56
3 Mathematics of the Simplex Method 61
3.1 Introduction 61
3.2 Some Basic Definitions 61
3.3 Some Elementary Results for LPP 64
3.4 The Simplex Algorithm: Main Theorems 70
3.5 A Useful Observation 80
3.6 The Revised Simplex Method 82
3.7 Summary and Additional Notes 93
3.8 Exercises 94
4 Duality in Linear Programming 101
4.1 Introduction 101
4.2 The Dual Problem: Motivation Through an Example 101
4.3 Construction of the Dual 103
4.4 Duality Theorems 107
4.5 A Convenient Way for Reading the Solution of the Dual 114
4.6 Some Useful Deductions from the Complementary Slackness Theorem 118
4.7 The Dual Simplex Method 120
4.8 Post Optimality Analysis 125
4.9 Two Person Zero-Sum Matrix Games 133
x Contents
4.10 Linear Programming and Matrix Game Equivalence 136
4.11 Summary and Additional Notes 143
4.12 Exercises 144
5 The Transportation and Assignment Problems 151
5.1 Introduction 151
5.2 The Transportation Problem : Description and Mathematical Model 151
5.3 The Balanced Transportation Problem 153
5.4 Consequence of Unimodularity 157
5.5 Optimality Conditions/ Stopping Criterion 161
5.6 Methods for Finding a Starting Solution 162
5.7 The Complete Algorithm 169
5.8 Degeneracy in the Transportation Problem 180
5.9 The Unbalanced Transportation Problem 183
5.10 The Assignment Problem: Description and Mathematical Model 190
5.11 A Transportation Equivalent of the Assignment Problem 194
5.12 An Important Lemma 197
5.13 The Hungarian Method for solving the Assignment Problem 198
5.14 A Dual Interpretation of the Hungarian Method 203
5.15 Finite Convergence of the Hungarian Method 207
5.16 Summary and Additional Notes 208
5.17 Exercises 210
6 Integer Linear Programming 217
6.1 Introduction 217
6.2 Mathematical Model and Some Possible Approaches 217
6.3 Gomory s Cutting Plane Method for All Integer Linear Programming 221
6.4 Gomory s Cutting Plane Method for Mixed Integer Linear Programming 230
6.5 Branch and Bound Method 236
6.6 Some Other Related Problems 241
6.7 Zero-One Implicit Enumeration Algorithm 244
6.8 Summary and Additional Notes 249
6.9 Exercises 249
7 Convex Optimization and Quadratic Programming 255
7.1 Introduction 255
7.2 Convex Functions and their Properties 255
7.3 Convex Optimization Problems 267
7.4 Convex Programming Problems 270
7.5 Optimality Conditions: Motivations from Elementary Calculus 274
7.6 Quadratic Programming 280
7.7 Wolfe s Method for Quadratic Programming 281
7.8 Summary and Additional Notes 292
7.9 Exercises 293
Contents xi
8 Optimality Conditions and Duality in Nonlinear Programming 299
8.1 Introduction 299
8.2 Feasible Directions and Linearizing Cone 299
8.3 Basic Constraint Qualification 306
8.4 Lagrangian and Lagrange Multipliers 308
8.5 Karush-Kuhn-Tucker Necessary/Sufficient Optimality Conditions 310
8.6 Duality in Nonlinear Programming 314
8.7 Certain Special Cases of Wolfe Dual 317
8.8 Summary and Additional Notes 320
8.9 Exercises 321
9 Unconstrained Optimization Problems 325
9.1 Introduction 325
9.2 Basic Scheme and Certain Desirable Properties 325
9.3 Line Search Methods for Unimodal Functions 327
9.4 The Steepest Descent Method 336
9.5 Newton s Method 341
9.6 Modified Newton s Method 343
9.7 The Conjugate Gradient Method 344
9.8 Davidon-Fletcher-Powell (DFP) Method 353
9.9 Preconditioning 356
9.10 Summary and Additional Notes 356
9.11 Exercises 357
10 Algorithms in Nonlinear Programming 363
10.1 Introduction 363
10.2 Frank and Wolfe s Method 364
10.3 Mathematical Justification of Frank and Wolfe s Method 368
10.4 Rosen s Gradient Projection Method 369
10.5 The Penalty Function Method: Motivation 374
10.6 Penalty Function Method: Some Illustrative Examples 378
10.7 Numerical Implementation of the Penalty Function Method 382
10.8 Mathematical Justification of the Penalty Function Method 384
10.9 The Barrier Function Method: Motivation 386
10.10 Analytical Implementation of the Barrier Function Method 387
10.11 Numerical Implementation of the Barrier Function Method 389
10.12 Mathematical Justification of the Barrier Function Method 391
10.13 Starting Point for the Barrier Function Method 392
10.14 Algorithms and Their Global Convergence 393
10.15 Multistage Decision Problems and Dynamic Programming 395
10.16 Summary and Additional Notes 403
10.17 Exercises 405
11 Computational Complexity and Karmarkar s Algorithm for
Linear Programming 409
11.1 Introduction 409
11.2 Simplex Algorithm: A Eeview of the Basic Facts 410
xii Contents
11.3 Computational Complexity of the Simplex Algorithm: Definitions 411
11.4 Simplex Algorithm is not a Polynomial Time Algorithm 413
11.5 Karmarkar s Algorithm for Linear Programming 419
11.6 Centering Transformation and Projection Matrix 421
11.7 The Complete Algorithm 426
11.8 Putting a general LPP in Karmarkar s form 430
11.9 Worst Case Computational Complexity of Karmarkar s Algorithm 433
11.10 Summary and Additional Notes 435
11.11 Exercises 437
12 Some Generalized Convex Functions and Fractional Programming 441
12.1 Introduction 441
12.2 Quasiconvex and Quasiconcave Functions 441
12.3 Pseudoconvex and Pseudoconcave Functions 454
12.4 Fractional Programming Problems 461
12.5 Linear Fractional Programming Problems 462
12.6 Nonlinear Fractional Programming Problems 468
12.7 Summary and Additional Notes 474
12.8 Exercises 475
13 Multi-objective Optimization: Theory and Methods 479
13.1 Introduction 479
13.2 Conflicting Objectives and Tradeoff 480
13.3 Various Solution Concepts 481
13.4 Weighted Sum Approach 489
13.5 Formulation of Goal Programming Problem 495
13.6 Solution Methodologies for Linear Goal Programming Problems 499
13.7 Summary and Additional Notes 504
13.8 Exercises 505
14 Semi-definite Programming 509
14.1 Introduction 509
14.2 Motivation 509
14.3 Cone 512
14.4 Formulation of Semi-definite Programming Problem 515
14.5 Applications 518
14.6 Formulation of the Dual Problem 523
14.7 Duality Theorems 527
14.8 Summary and Additional Notes 532
14.9 Exercises 533
15 Evolutionary Methods and Global Optimization 535
15.1 Introduction 535
15.2 How Difficult is the Problem? 536
15.3 Simulated Annealing 538
15.4 Genetic Algorithms 542
15.5 Ant Colony Optimization 547
15.6 Summary and Additional Notes 551
Contents xiii
16 Mathematical Programming Applications in Machine Learning 553
16.1 Introduction 553
16.2 Binary (Two-Class) Pattern Classification Problems 554
16.3 Optimal Separation For Linearly Separable Data Sets 560
16.4 Hard Margin Classifier 563
16.5 Soft Margin Classifier 568
16.6 Nonlinear Support Vector Machines 572
16.7 Summary and Additional Notes 576
16.8 Exercises 577
17 Mathematical Programming Applications in Financial Mathematics 579
17.1 Introduction 579
17.2 Technical Terminologies 579
17.3 Two Asset Portfolio Optimization 583
17.4 Multi Asset Portfolio Optimization 589
17.5 Capital Asset Pricing Model (CAPM) 596
17.6 Summary and Additional Notes 603
17.7 Exercises 605
18 Mathematical Programming Applications in Engineering 609
18.1 Introduction 609
18.2 Optimization in VLSI Design 609
18.3 Global Routing for VLSI Standard Cells 612
18.4 Wire-Sizing and Buffer Sizing 613
18.5 Synthesis of Antennae Array 616
18.6 Chemical Equilibrium 618
18.7 Protein Folding 619
18.8 Curve Fitting 621
18.9 Reliability Optimization 622
18.10 Wireless Network System 624
18.11 NMR Experiment Design 625
18.12 Summary and Additional Notes 627
19 Matlab Codes For Some Selected Algorithms 629
19.1 Introduction 629
19.2 Simplex Algorithm 632
19.3 Dual Simplex Algorithm 650
19.4 Assignment Problem: Hungarian Method 655
19.5 Branch and Bound Method 659
19.6 Golden Section Rule 671
19.7 Steepest Descent Method 673
19.8 Conjugate Gradient Method 675
19.9 DFP Method 676
19.10 Frank and Wolfe s Method 678
19.11 General Comments 680
References 681
Index 691
|
| adam_txt |
Titel: Numerical optimization with applications
Autor: Chandra, Suresh
Jahr: 2009
Contents
Preface vii
1 Introduction 1
1.1 An Overview of Optimization 1
1.2 Optimization Problems 2
1.3 Some Simple Illustrative Examples 5
2 Linear Programming 11
2.1 Introduction 11
2.2 The Graphical Method and Some Intuitive Results 11
2.3 The Simplex Method as an Algebraic Version of The Graphical Method 16
2.4 The Simplex Method: Certain Notations 24
2.5 Description of the Simplex Method 26
2.6 Methods of Artificial Variables 33
2.7 Alternate Optima 44
2.8 Redundancy in Linear Programming 46
2.9 Degeneracy and Cycling 51
2.10 The Simplex Tableau in the Condensed Form 53
2.11 Summary and Additional Notes 55
2.12 Exercises 56
3 Mathematics of the Simplex Method 61
3.1 Introduction 61
3.2 Some Basic Definitions 61
3.3 Some Elementary Results for LPP 64
3.4 The Simplex Algorithm: Main Theorems 70
3.5 A Useful Observation 80
3.6 The Revised Simplex Method 82
3.7 Summary and Additional Notes 93
3.8 Exercises 94
4 Duality in Linear Programming 101
4.1 Introduction 101
4.2 The Dual Problem: Motivation Through an Example 101
4.3 Construction of the Dual 103
4.4 Duality Theorems 107
4.5 A Convenient Way for Reading the Solution of the Dual 114
4.6 Some Useful Deductions from the Complementary Slackness Theorem 118
4.7 The Dual Simplex Method 120
4.8 Post Optimality Analysis 125
4.9 Two Person Zero-Sum Matrix Games 133
x Contents
4.10 Linear Programming and Matrix Game Equivalence 136
4.11 Summary and Additional Notes 143
4.12 Exercises 144
5 The Transportation and Assignment Problems 151
5.1 Introduction 151
5.2 The Transportation Problem : Description and Mathematical Model 151
5.3 The Balanced Transportation Problem 153
5.4 Consequence of Unimodularity 157
5.5 Optimality Conditions/ Stopping Criterion 161
5.6 Methods for Finding a Starting Solution 162
5.7 The Complete Algorithm 169
5.8 Degeneracy in the Transportation Problem 180
5.9 The Unbalanced Transportation Problem 183
5.10 The Assignment Problem: Description and Mathematical Model 190
5.11 A Transportation Equivalent of the Assignment Problem 194
5.12 An Important Lemma 197
5.13 The Hungarian Method for solving the Assignment Problem 198
5.14 A Dual Interpretation of the Hungarian Method 203
5.15 Finite Convergence of the Hungarian Method 207
5.16 Summary and Additional Notes 208
5.17 Exercises 210
6 Integer Linear Programming 217
6.1 Introduction 217
6.2 Mathematical Model and Some Possible Approaches 217
6.3 Gomory's Cutting Plane Method for All Integer Linear Programming 221
6.4 Gomory's Cutting Plane Method for Mixed Integer Linear Programming 230
6.5 Branch and Bound Method 236
6.6 Some Other Related Problems 241
6.7 Zero-One Implicit Enumeration Algorithm 244
6.8 Summary and Additional Notes 249
6.9 Exercises 249
7 Convex Optimization and Quadratic Programming 255
7.1 Introduction 255
7.2 Convex Functions and their Properties 255
7.3 Convex Optimization Problems 267
7.4 Convex Programming Problems 270
7.5 Optimality Conditions: Motivations from Elementary Calculus 274
7.6 Quadratic Programming 280
7.7 Wolfe's Method for Quadratic Programming 281
7.8 Summary and Additional Notes 292
7.9 Exercises 293
Contents xi
8 Optimality Conditions and Duality in Nonlinear Programming 299
8.1 Introduction 299
8.2 Feasible Directions and Linearizing Cone 299
8.3 Basic Constraint Qualification 306
8.4 Lagrangian and Lagrange Multipliers 308
8.5 Karush-Kuhn-Tucker Necessary/Sufficient Optimality Conditions 310
8.6 Duality in Nonlinear Programming 314
8.7 Certain Special Cases of Wolfe Dual 317
8.8 Summary and Additional Notes 320
8.9 Exercises 321
9 Unconstrained Optimization Problems 325
9.1 Introduction 325
9.2 Basic Scheme and Certain Desirable Properties 325
9.3 Line Search Methods for Unimodal Functions 327
9.4 The Steepest Descent Method 336
9.5 Newton's Method 341
9.6 Modified Newton's Method 343
9.7 The Conjugate Gradient Method 344
9.8 Davidon-Fletcher-Powell (DFP) Method 353
9.9 Preconditioning 356
9.10 Summary and Additional Notes 356
9.11 Exercises 357
10 Algorithms in Nonlinear Programming 363
10.1 Introduction 363
10.2 Frank and Wolfe's Method 364
10.3 Mathematical Justification of Frank and Wolfe's Method 368
10.4 Rosen's Gradient Projection Method 369
10.5 The Penalty Function Method: Motivation 374
10.6 Penalty Function Method: Some Illustrative Examples 378
10.7 Numerical Implementation of the Penalty Function Method 382
10.8 Mathematical Justification of the Penalty Function Method 384
10.9 The Barrier Function Method: Motivation 386
10.10 Analytical Implementation of the Barrier Function Method 387
10.11 Numerical Implementation of the Barrier Function Method 389
10.12 Mathematical Justification of the Barrier Function Method 391
10.13 Starting Point for the Barrier Function Method 392
10.14 Algorithms and Their Global Convergence 393
10.15 Multistage Decision Problems and Dynamic Programming 395
10.16 Summary and Additional Notes 403
10.17 Exercises 405
11 Computational Complexity and Karmarkar's Algorithm for
Linear Programming 409
11.1 Introduction 409
11.2 Simplex Algorithm: A Eeview of the Basic Facts 410
xii Contents
11.3 Computational Complexity of the Simplex Algorithm: Definitions 411
11.4 Simplex Algorithm is not a Polynomial Time Algorithm 413
11.5 Karmarkar's Algorithm for Linear Programming 419
11.6 Centering Transformation and Projection Matrix 421
11.7 The Complete Algorithm 426
11.8 Putting a general LPP in Karmarkar's form 430
11.9 Worst Case Computational Complexity of Karmarkar's Algorithm 433
11.10 Summary and Additional Notes 435
11.11 Exercises 437
12 Some Generalized Convex Functions and Fractional Programming 441
12.1 Introduction 441
12.2 Quasiconvex and Quasiconcave Functions 441
12.3 Pseudoconvex and Pseudoconcave Functions 454
12.4 Fractional Programming Problems 461
12.5 Linear Fractional Programming Problems 462
12.6 Nonlinear Fractional Programming Problems 468
12.7 Summary and Additional Notes 474
12.8 Exercises 475
13 Multi-objective Optimization: Theory and Methods 479
13.1 Introduction 479
13.2 Conflicting Objectives and Tradeoff 480
13.3 Various Solution Concepts 481
13.4 Weighted Sum Approach 489
13.5 Formulation of Goal Programming Problem 495
13.6 Solution Methodologies for Linear Goal Programming Problems 499
13.7 Summary and Additional Notes 504
13.8 Exercises 505
14 Semi-definite Programming 509
14.1 Introduction 509
14.2 Motivation 509
14.3 Cone 512
14.4 Formulation of Semi-definite Programming Problem 515
14.5 Applications 518
14.6 Formulation of the Dual Problem 523
14.7 Duality Theorems 527
14.8 Summary and Additional Notes 532
14.9 Exercises 533
15 Evolutionary Methods and Global Optimization 535
15.1 Introduction 535
15.2 How Difficult is the Problem? 536
15.3 Simulated Annealing 538
15.4 Genetic Algorithms 542
15.5 Ant Colony Optimization 547
15.6 Summary and Additional Notes 551
Contents xiii
16 Mathematical Programming Applications in Machine Learning 553
16.1 Introduction 553
16.2 Binary (Two-Class) Pattern Classification Problems 554
16.3 Optimal Separation For Linearly Separable Data Sets 560
16.4 Hard Margin Classifier 563
16.5 Soft Margin Classifier 568
16.6 Nonlinear Support Vector Machines 572
16.7 Summary and Additional Notes 576
16.8 Exercises 577
17 Mathematical Programming Applications in Financial Mathematics 579
17.1 Introduction 579
17.2 Technical Terminologies 579
17.3 Two Asset Portfolio Optimization 583
17.4 Multi Asset Portfolio Optimization 589
17.5 Capital Asset Pricing Model (CAPM) 596
17.6 Summary and Additional Notes 603
17.7 Exercises 605
18 Mathematical Programming Applications in Engineering 609
18.1 Introduction 609
18.2 Optimization in VLSI Design 609
18.3 Global Routing for VLSI Standard Cells 612
18.4 Wire-Sizing and Buffer Sizing 613
18.5 Synthesis of Antennae Array 616
18.6 Chemical Equilibrium 618
18.7 Protein Folding 619
18.8 Curve Fitting 621
18.9 Reliability Optimization 622
18.10 Wireless Network System 624
18.11 NMR Experiment Design 625
18.12 Summary and Additional Notes 627
19 Matlab Codes For Some Selected Algorithms 629
19.1 Introduction 629
19.2 Simplex Algorithm 632
19.3 Dual Simplex Algorithm 650
19.4 Assignment Problem: Hungarian Method 655
19.5 Branch and Bound Method 659
19.6 Golden Section Rule 671
19.7 Steepest Descent Method 673
19.8 Conjugate Gradient Method 675
19.9 DFP Method 676
19.10 Frank and Wolfe's Method 678
19.11 General Comments 680
References 681
Index 691 |
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| author | Chandra, Suresh 1944- Jayadeva Mehra, Aparna |
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| oclc_num | 228375454 |
| open_access_boolean | |
| owner | DE-20 DE-11 DE-703 DE-29T |
| owner_facet | DE-20 DE-11 DE-703 DE-29T |
| physical | XIII, 694 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Alpha Science Internat. |
| record_format | marc |
| spelling | Chandra, Suresh 1944- Verfasser (DE-588)142013692 aut Numerical optimization with applications Suresh Chandra ; Jayadeva ; Aparna Mehra Oxford Alpha Science Internat. 2009 XIII, 694 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical optimization Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Optimierung (DE-588)4043664-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Jayadeva Verfasser aut Mehra, Aparna Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016783187&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chandra, Suresh 1944- Jayadeva Mehra, Aparna Numerical optimization with applications Mathematical optimization Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4043664-0 |
| title | Numerical optimization with applications |
| title_auth | Numerical optimization with applications |
| title_exact_search | Numerical optimization with applications |
| title_exact_search_txtP | Numerical optimization with applications |
| title_full | Numerical optimization with applications Suresh Chandra ; Jayadeva ; Aparna Mehra |
| title_fullStr | Numerical optimization with applications Suresh Chandra ; Jayadeva ; Aparna Mehra |
| title_full_unstemmed | Numerical optimization with applications Suresh Chandra ; Jayadeva ; Aparna Mehra |
| title_short | Numerical optimization with applications |
| title_sort | numerical optimization with applications |
| topic | Mathematical optimization Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Mathematical optimization Numerisches Verfahren Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016783187&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT chandrasuresh numericaloptimizationwithapplications AT jayadeva numericaloptimizationwithapplications AT mehraaparna numericaloptimizationwithapplications |