Convexity and optimization in finite dimensions I

Contents......Page vii
Introduction......Page 1
1.1. Linear Combinations of Inequalities......Page 7
1.2. Fourier Elimination......Page 11
1.3. Proof of the Kuhn-Fourier Theorem......Page 15
1.4. Consequence Relations. The Farkas Lemma......Page 17
1.5. Irreducibly Inconsistent Systems......Page 20
1.6. Transposition Theorems......Page 23
1.7. The Duality Theorem of Linear Programming......Page 26
2.1. Means and Averages......Page 31
2.2. Dimensions......Page 33
2.3. Polyhedra and their Boundaries......Page 36
2.4. Extreme and Exposed Sets......Page 39
2.5. Primitive Faces. The Finite Basis Theorem......Page 43
2.6. Subspaces. Orthogonality......Page 47
2.7. Cones. Polarity......Page 51
2.8. Polyhedral Cones......Page 55
2.9. A Direct Proof of the Theorem of Weyl......Page 57
2.10. Lineality Spaces......Page 59
2.11. Homogenization......Page 62
2.12. Decomposition and Separation of Polyhedra......Page 65
2.13. Face Lattices of Polyhedral Cones......Page 69
2.14. Polar and Dual Polyhedra......Page 73
2.15. Gale Diagrams......Page 78
3.1. The Normed Linear Space R^n......Page 82
3.2. Closure and Relative Interior of Convex Sets......Page 87
3.3. Separation of Convex Sets......Page 95
3.4. Supporting Planes and Cones......Page 100
3.5. Boundedness and Polarity......Page 104
3.6. Extremal Properties......Page 109
3.7. Combinatorial Properties......Page 117
3.8. Topological Properties......Page 123
3.9. Fixed Point Theorems......Page 126
3.10. Norms and Support Functions......Page 131
4.1. Convex Functions......Page 134
4.2. Epigraphs......Page 140
4.3. Directorial Derivatives......Page 143
4.4. Differentiable Convex Functions......Page 149
4.5. A Regularity Condition......Page 153
4.6. Conjugate Functions......Page 156
4.7. Strongly Closed Convex Functions......Page 159
4.8. Examples of Conjugate Functions......Page 161
4.9. Generalization of Convexity......Page 169
4.10. Pseudolinear Functions......Page 173
5.1. The Duality Theorem of Fenchel......Page 177
5.2. Duality Gaps......Page 181
5.3. Generalization of Fenchel's Duality Theorem......Page 186
5.4. Proof of the Generalized Fenchel Theorem......Page 191
5.5. Alternative Characterizations of Stability......Page 197
5.6. Generation of Stable Functions......Page 202
5.7. Rockafellar's Duality Theorem......Page 204
5.8. Duality Theorems of the Dennis-Dorn Type......Page 208
5.9. Duality Theorems for Quadratic Programs......Page 216
6.1. The Minimax Theorem of v. Neumann......Page 221
6.2. Saddle Points......Page 226
6.3. Minimax Theorems for Compact Sets......Page 230
6.4. Minimax Theorems for Noncompact Sets......Page 234
6.5. Lagrange Multipliers......Page 240
6.6. Kuhn-Tucker Theory for Differentiable Functions......Page 246
6.7. Saddle Points of the Lagrangian......Page 248
6.8. Duality Theorems and Lagrange Multipliers......Page 254
6.9. Constrained Minimax Programs......Page 256
6.10. Systems of Convex Inequalities......Page 263
Bibliography......Page 269
Author and Subject Index......Page 286