Author | Stoer, Josef. author |
---|---|
Title | Convexity and Optimization in Finite Dimensions I [electronic resource] / by Josef Stoer, Christoph Witzgall |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1970 |
Connect to | http://dx.doi.org/10.1007/978-3-642-46216-0 |
Descript | X, 298 p. online resource |
1 Inequality Systems -- 1.1. Linear Combinations of Inequalities -- 1.2. Fourier Elimination -- 1.3. Proof of the Kuhn-Fourier Theorem -- 1.4. Consequence Relations. The Farkas Lemma -- 1.5. Irreducibly Inconsistent Systems -- 1.6. Transposition Theorems -- 1.7. The Duality Theorem of Linear Programming -- 2 Convex Polyhedra -- 2.1. Means and Averages -- 2.2. Dimensions -- 2.3. Polyhedra and their Boundaries -- 2.4. Extreme and Exposed Sets -- 2.5. Primitive Faces. The Finite Basis Theorem -- 2.6. Subspaces. Orthogonality -- 2.7. Cones. Polarity -- 2.8. Polyhedral Cones -- 2.9. A Direct Proof of the Theorem of Weyl -- 2.10. Lineality Spaces -- 2.11. Homogenization -- 2.12. Decomposition and Separation of Polyhedra -- 2.13. Face Lattices of Polyhedral Cones -- 2.14. Polar and Dual Polyhedra -- 2.15. Gale Diagrams -- 3 Convex Sets -- 3.1. The Normed Linear Space Rn -- 3.2. Closure and Relative Interior of Convex Sets -- 3.3. Separation of Convex Sets -- 3.4. Supporting Planes and Cones -- 3.5. Boundedness and Polarity -- 3.6. Extremal Properties -- 3.7. Combinatorial Properties -- 3.8. Topological Properties -- 3.9. Fixed Point Theorems -- 3.10. Norms and Support Functions -- 4 Convex Functions -- 4.1. Convex Functions -- 4.2. Epigraphs -- 4.3. Directorial Derivatives -- 4.4. Differentiable Convex Functions -- 4.5. A Regularity Condition -- 4.6. Conjugate Functions -- 4.7. Strongly Closed Convex Functions -- 4.8. Examples of Conjugate Functions -- 4.9. Generalization of Convexity -- 4.10. Pseudolinear Functions -- 5 Duality Theorems -- 5.1. The Duality Theorem of Fenchel -- 5.2. Duality Gaps -- 5.3. Generalization of Fenchelโs Duality Theorem -- 5.4. Proof of the Generalized Fenchel Theorem -- 5.5. Alternative Characterizations of Stability -- 5.6. Generation of Stable Functions -- 5.7. Rockafellarโs Duality Theorem -- 5.8. Duality Theorems of the Dennis-Dorn Type -- 5.9. Duality Theorems for Quadratic Programs -- 6 Saddle Point Theorems -- 6.1. The Minimax Theorem of v. Neumann -- 6.2. Saddle Points -- 6.3. Minimax Theorems for Compact Sets -- 6.4. Minimax Theorems for Noncompact Sets -- 6.5. Lagrange Multipliers -- 6.6. Kuhn-Tucker Theory for Differentiable Functions -- 6.7. Saddle Points of the Lagrangian -- 6.8. Duality Theorems and Lagrange Multipliers -- 6.9. Constrained Minimax Programs -- 6.10. Systems of Convex Inequalities -- Author and Subject Index