Optimization theory: the finite dimensional case

PREFACE ......Page 5
CONTENTS ......Page 9
1. Preliminary remarks ......Page 15
2. Existence of extreme points ......Page 16
3. Unconstrained local minima and maxima ......Page 25
4. Positive symmetric matrices ......Page 35
5. Minima of convex functions ......Page 41
6. Linearly constrained local minima and maxima ......Page 51
7. Minima of quadratic functions ......Page 60
8. Inner product spaces and the Gram-Schmidt process ......Page 71
9. Eigenvalues and eigenvectors of a real symmetric matrix ......Page 84
2. Relative eigenvalues and eigenvectors ......Page 99
3. Eigenvalues on subspaces ......Page 108
4. A fundamental lemma ......Page 111
5. Quadratic forms of arcs ......Page 115
6. Further properties of quadratic forms ......Page 126
7. Rectangular matrices, norms, and principal values ......Page 132
8. Pseudoinverses and *-reciprocals of matrices ......Page 144
1. General remarks ......Page 152
2. Constrained minima ......Page 153
3. Sufficient conditions for a local constrained minimum ......Page 165
4. Classification of critical points ......Page 170
5. Tangent vectors and normality ......Page 175
6. Gradient of a function relative to constraints ......Page 181
7. Implicit function theorems ......Page 185
1. Introduction ......Page 191
2. The Lagrange multiplier rule ......Page 193
3. Convex cones and their duals ......Page 201
4. Tangent cones and normal cones ......Page 217
5. Remarks on differentiation ......Page 224
6. Minimum of a function on an arbitrary set S ......Page 227
7. Minima subject to nonlinear constraints ......Page 235
8. Concave and linear constraints ......Page 244
9. Linear programming ......Page 248
10. Criteria for regularity ......Page 251
11. Infinitely many inequality constraints ......Page 259
12. A simple integral problem ......Page 263
1. Introduction ......Page 267
2. Penalty functions ......Page 269
3. Removal of constraints ......Page 277
4. Augmentability and the Lagrange multiplier rule ......Page 281
5. Minima subject to simple inequality constraints ......Page 287
6. Augmentability in the case of inequality constraints ......Page 295
7. An extended Lagrange multiplier rule ......Page 303
8. Quasinormality ......Page 310
9. Extensions of earlier results ......Page 316
10. A method of multipliers ......Page 321
1. Introduction ......Page 327
2. Illustrative examples ......Page 329
3. Convex programming ......Page 338
4. Convex programming with linear constraints ......Page 349
5. Further examples ......Page 357
6. Minima relative to general inequality constraints ......Page 361
7. Applications to quadratic forms ......Page 368
8. Implicit function theorems ......Page 375
9. Derived cones and derived sets ......Page 378
10. Minima relative to equality and inequality constraints ......Page 385
11. Further theorems on tangent spaces and derived sets ......Page 398
12. Auxiliary theorems ......Page 402
2. Euclidean $n$-space ......Page 407
3. Basic concepts ......Page 420
4. Extreme points of functions ......Page 425
5. Local minima and maxima—the one-variable case ......Page 430
6. Functions of class $C^{(k)}$ ......Page 441
BIBLIOGRAPHY ......Page 451
INDEX ......Page 459