线性代数及其在规划中的应用【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

线性代数及其在规划中的应用PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Part Ⅰ3

Chapter 1 Background and Fundamentals of Mathematics3

1.1 Basic Concepts3

1.2 Relations4

1.3 Functions5

1.4 The Integers9

1.4.1 Long Division9

1.4.2 Relatively Prime10

1.4.3 Prime10

1.4.4 The Unique Factorization Theorem11

Chapter 2 Groups13

2.1 Groups13

2.2 Subgroups15

2.3 Normal Subgroups17

2.4 Homomorphisms19

2.5 Permutations20

2.6 Product of Groups22

Chapter 3 Rings24

3.1 Commutative Rings24

3.2 Units25

3.3 The Integers Mod N26

3.4 Ideals and Quotient Rings27

3.5 Homomorphism28

3.6 Polynomial Rings29

3.6.1 The Division Algorithm30

3.6.2 Associate31

3.7 Product of Rings32

3.8 Characteristic33

3.9 Boolean Rings34

Chapter 4 Matrices and Matrix Rings35

4.1 Elementary Operations and Elementary Matrices35

4.2 Systems of Equations36

4.3 Determinants37

4.4 Similarity40

Part Ⅱ45

Chapter 5 Vector Spaces45

5.1 The Axioms for a Vector Space45

5.2 Linear Independence,Dimension,and Basis48

5.3 Intersection,Sum and Direct Sum of Subspaces53

5.4 Factor Space56

5.5 Inner Product Spaces58

5.6 Orthonormal Bases and Orthogonal Complements62

5.7 Reciprocal Basis and Change of Basis65

Chapter 6 Linear Transformations71

6.1 Definition of Linear Transformation71

6.2 Sums and Products of Liner Transformations75

6.3 Special Types of Linear Transformations77

6.4 The Adjoint of a Linear Transformation82

6.5 Component Formulas89

Chapter 7 Determinants And Matrices92

7.1 The Generalized Kronecker Deltas and the Summation Convention92

7.2 Determinants96

7.3 The Matrix of a Linear Transformation99

7.4 Solution of Systems of Linear Equation102

7.5 Special Matrices103

Chapter 8 Spectral Decompositions108

8.1 Direct Sum of Endomorphisms108

8.2 Eigenvectors and Eigenvalues109

8.3 The Characteristic Polynomial110

8.4 Spectral Decomposition for Hermitian Endomorphisms113

8.5 Illustrative Examples124

8.6 The Minimal Polynomial127

8.7 Spectral Decomposition for Arbitrary Endomorphisms130

Chapter 9 Tensor Algebra144

9.1 Linear Functions,the Dual Space144

9.2 The Second Dual Space,Canonical Isomorphisms149

Part Ⅲ157

Chapter 10 Linear Programming157

10.1 Basic Properties of Linear Programs157

10.2 Many Computational Procedures to Simplex Method162

10.3 Duality170

10.3.1 Dual Linear Programs170

10.3.2 The Duality Theorem172

10.3.3 Relations to the Simplex Procedure173

10.4 Interior-point Methods176

10.4.1 Elements of Complexity Theory176

10.4.2 The Analytic Center176

10.4.3 The Central Path177

10.4.4 Solution Strategies180

Chapter 11 Unconstrained Problems187

11.1 Transportation and Network Flow Problems187

11.1.1 The Transportation Problem187

11.1.2 The Northwest Corner Rule190

11.1.3 Basic Network Concepts190

11.1.4 Maximal Flow191

11.2 Basic Properties of Solutions and Algorithms197

11.2.1 First-order Necessary Conditions197

11.2.2 Second-order Conditions198

11.2.3 Minimization and Maximization of Convex Functions198

11.2.4 Zeroth-order Conditions198

11.2.5 Global Convergence of Descent Algorithms199

11.2.6 Speed of Convergence202

11.3 Basic Descent Methods204

11.3.1 Fibonacci and Golden Section Search204

11.3.2 Closedness of Line Search Mgorithms205

11.3.3 Line Search207

11.3.4 The Steepest Descent Method209

11.3.5 Coordinate Descent Methods211

11.4 Conjugate Direction Methods212

11.4.1 Conjugate Directions212

11.4.2 Descent Properties of the Conjugate Direction Method214

11.4.3 The Conjugate Gradient Method214

11.4.4 The C-G Method as an Optimal Process215

Chapter 12 Constrained Minimization218

12.1 Quasi-Newton Methods218

12.1.1 Modified Newton Method218

12.1.2 Scaling219

12.1.3 Memoryless Quasi-Newton Methods222

12.2 Constrained Minimization Conditions223

12.2.1 Constraints223

12.2.2 Tangent Plane224

12.2.3 First-order Necessary Conditions(Equality Constraints)224

12.2.4 Second-order Conditions225

12.2.5 Eigenvalues in Tangent Subspace225

12.2.6 Inequality Constraints227

12.2.7 Zeroth-order Conditions and Lagrange Multipliers228

12.3 Primal Methods231

12.3.1 Feasible Direction Methods231

12.3.2 Active Set Methods231

12.3.3 The Gradient Projection Method232

12.3.4 Convergence Rate of the Gradient Projection Method233

12.3.5 The Reduced Gradient Method235

12.4 Penalty and Barrier Methods237

12.4.1 Penalty Methods237

12.4.2 Barrier Methods238

12.4.3 Properties of Penalty and Barrier Functions239

12.5 Dual and Cutting Plane Methods243

12.5.1 Global Duality243

12.5.2 Local Duality245

12.5.3 Dual Canonical Convergence Rate247

12.5.4 Separable Problems248

12.5.5 Decomposition248

12.5.6 The Dual Viewpoint249

12.5.7 Cutting Plane Methods251

12.5.8 Kelley's Convex Cutting Plane Algorithm253

12.5.9 Modifications253

12.6 Primal-dual Methods254

12.6.1 The Standard Problem254

12.6.2 Strategies256

12.6.3 A Simple Merit Function257

12 6 4 Basic Primal-dual Methods257

12.6.5 Modified Newton Methods260

12.6.6 Descent Properties262

12.6.7 Interior Point Methods263

Bibliography267