Chapter 1 Introduction
1.1 Problem Statement and Basic Definitions 2
1.2 Some Illustrative Examples 4
Exercises 24
Notes and References 28
Part 1 Convex Analysis
Chapter 2 Convex Sets
2.1 Convex Hulls 34
2.2 Closure and Interior of a Convex Set 34
2.3 Separation and Support of Convex Sets 41
2.4 Convex Cones and Polarity 51
2.5 Polyhefral Sets, Extreme Points, and Extreme Directions 53
2.6 Linear Programming and the Simplex Method 63
Exercises 72
Notes and References 77
Chapter 3 Convex Functions
3.1 Definitions and Baisc Properties 80
3.2 Subgradients of Convex FUnctions 84
3.3 Differentiable Convex Functions 89
3.4 Minima and Maxima of Convex Functions 94
3.5 Generalizations of Convex Functions 99
Exercises 112
Notes and References 119
Part 2 Optimality COnditions and Duality
Chapter 4 The Fritz John and the Kuhn-Tucker Optimality Conditions
4.1 Unconstrained Problems 124
4.2 Problems with Inequality Constraints 127
4.3 Problems with Inequality and Equality Constraints 140
Exercises 151
Notes and References 159
Chapter 5 Constraint Qualifications
5.1 The Cone of Tangents 161
5.2 Other Consistent Qualifications 165
5.3 Problems with Inequality and Equality Constraints 167
Exercises 172
Notes and References 174
Chapter 6 Lagrangian Duality and Saddle Point Optimality Conditions
6.1 The Lagrangian Dual Problem 176
6.2 Duality Theorems and Saddle Point Optimality Conditions 180
6.3 Properties of the Dual Function 187
6.4 Solving the Dual Problem 196
6.5 Getting the Primal Solution 207
6.6 Linear and Quadratic Programs 211
Exercises 215
Notes and References 224
Part 3 Algorithms and Their Convergence
Chapter 7 The Concept of an Algorithm
7.1 Algorithms and Algorithmic Maps 229
7.2 Closed Maps and Convergence 231
7.3 Composition of Mappings 236
7.4 Comparison Among Algorithms 241
Exercises 244
Notes and References 251
Chapter 8 Unconstrained Optimization
8.1 Line Search Without Using Derivatives 253
8.2 Line Search Using Derivatives 264
8.3 Closedness of the Line Search Algorithmic Map 269
8.4 Multidimensional Search Without Using Derivatives 270
8.5 Multidimensional Search Using Derivatives 289
8.6 Methods Using Conjugate Directions 297
Exercises 317
Notes and References 327
Chapter 9 Penalty and barrier Functions
9.1 The COncept of Penalty Functions 332
9.2 Penalty Function Methods 336
9.3 Barrier Function Methods 342
Exercises 350
Notes and References 358
Chapter 10 Methods of Feasible Directions
10.1 The Method of Zoutendijk 361
10.2 Convergence Analysis of the Method of Zoutendijk 378
10.3 The Gradient Projection Method of Roson 389
10.4 The Method of Reduced Gradient of Wolfe 399
10.5 The Convex-Simplex Method of Zangwill 407
Exercises 416
Notes and References 434
Chapter 11 Linear Complementarity, and Quadratic, Separable, and Fractional Programming
11.1 The Linear Complementary Problem 438
11.2 Quadratic Programming 447
11.3 Separable Programming 453
11.4 Linear Fractinal Programming 471
Exercises 480
Notes and References 494
Appendix A Mathematical Review 497
Appendix B Summary of Convexity, Optimality Conditions, and Duality 505
Bibliography 516
Index 553
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