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付晓强

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[资源] linear and nonlinear optimization (second edition).pdf

Contents
Preface xiii
I Basics 1
1 Optimization Models 3
1.1 Introduction ............................... 3
1.2 Optimization: An Informal Introduction ................ 4
1.3 Linear Equations ............................. 7
1.4 Linear Optimization ........................... 10
Exercises ..................................... 12
1.5 Least-Squares Data Fitting ....................... 12
Exercises ..................................... 14
1.6 Nonlinear Optimization ......................... 14
1.7 Optimization Applications ........................ 18
1.7.1 Crew Scheduling and Fleet Scheduling .......... 18
Exercises ..................................... 22
1.7.2 Support Vector Machines ................. 22
Exercises ..................................... 24
1.7.3 Portfolio Optimization ................... 25
Exercises ..................................... 27
1.7.4 Intensity Modulated Radiation Treatment Planning .... 28
Exercises ..................................... 31
1.7.5 Positron Emission Tomography Image Reconstruction . . 32
Exercises ..................................... 34
1.7.6 Shape Optimization .................... 35
1.8 Notes ................................... 40
2 Fundamentals of Optimization 43
2.1 Introduction ............................... 43
2.2 Feasibility and Optimality ........................ 43
Exercises ..................................... 47
2.3 Convexity ................................ 48
2.3.1 Derivatives and Convexity ................. 50
Exercises ..................................... 52
2.4 The General Optimization Algorithm .................. 54
Exercises ..................................... 58
2.5 Rates of Convergence .......................... 58
Exercises ..................................... 61
2.6 Taylor Series ............................... 62
Exercises ..................................... 65
2.7 Newton’s Method for Nonlinear Equations ............... 67
2.7.1 Systems of Nonlinear Equations .............. 72
Exercises ..................................... 74
2.8 Notes ................................... 76
3 Representation of Linear Constraints 77
3.1 Basic Concepts .............................. 77
Exercises ..................................... 82
3.2 Null and Range Spaces ......................... 82
Exercises ..................................... 84
3.3 Generating Null-Space Matrices ..................... 86
3.3.1 Variable Reduction Method ................ 86
3.3.2 Orthogonal Projection Matrix ............... 89
3.3.3 Other Projections ..................... 90
3.3.4 The QR Factorization ................... 90
Exercises ..................................... 91
3.4 Notes ................................... 93
II Linear Programming 95
4 Geometry of Linear Programming 97
4.1 Introduction ............................... 97
Exercises ..................................... 98
4.2 Standard Form ..............................100
Exercises .....................................105
4.3 Basic Solutions and Extreme Points ...................106
Exercises .....................................114
4.4 Representation of Solutions; Optimality ................117
Exercises .....................................123
4.5 Notes ...................................124
5 The Simplex Method 125
5.1 Introduction ...............................125
5.2 The Simplex Method ..........................126
5.2.1 General Formulas .....................129
5.2.2 Unbounded Problems ...................134
5.2.3 Notation for the Simplex Method (Tableaus) .......135
5.2.4 Deficiencies of the Tableau ................139
5.3 The Simplex Method (Details) .....................144
5.3.1 Multiple Solutions .....................144
5.3.2 Feasible Directions and Edge Directions .........145
Exercises .....................................148
5.4 Getting Started—Artificial Variables ..................149
5.4.1 The Two-Phase Method ..................150
5.4.2 The Big-M Method ....................156
Exercises .....................................159
5.5 Degeneracy and Termination ......................162
5.5.1 Resolving Degeneracy Using Perturbation ........167
Exercises .....................................170
5.6 Notes ...................................171
6 Duality and Sensitivity 173
6.1 The Dual Problem ............................173
Exercises .....................................177
6.2 Duality Theory ..............................179
6.2.1 Complementary Slackness .................182
6.2.2 Interpretation of the Dual .................184
Exercises .....................................185
6.3 The Dual Simplex Method ........................189
Exercises .....................................194
6.4 Sensitivity ................................195
Exercises .....................................201
6.5 Parametric Linear Programming .....................204
Exercises .....................................210
6.6 Notes ...................................211
7 Enhancements of the Simplex Method 213
7.1 Introduction ...............................213
7.2 Problems with Upper Bounds ......................214
Exercises .....................................221
7.3 Column Generation ...........................222
Exercises .....................................227
7.4 The Decomposition Principle ......................227
Exercises .....................................238
7.5 Representation of the Basis .......................240
7.5.1 The Product Form of the Inverse .............240
7.5.2 Representation of the Basis—The LU Factorization . . . 248
Exercises .....................................256
7.6 Numerical Stability and Computational Efficiency ...........259
7.6.1 Pricing ...........................260
7.6.2 The Initial Basis ......................264
7.6.3 Tolerances; Degeneracy ..................265
7.6.4 Scaling ...........................266linear and nonlinear optimization (second edition).pdf
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