Convexity and Optimization in Finite Dimensions I

Dantzig's development of linear programming into one of the most applicable optimization techniques has spread interest in the algebra of linear inequalities, the geometry of polyhedra, the topology of convex sets, and the analysis of convex functions. It is the goal of this volume to provide a synopsis of these topics, and thereby the theoretical back ground for the arithmetic of convex optimization to be treated in a sub sequent volume. The exposition of each chapter is essentially independent, and attempts to reflect a specific style of mathematical reasoning. The emphasis lies on linear and convex duality theory, as initiated by Gale, Kuhn and Tucker, Fenchel, and v. Neumann, because it represents the theoretical development whose impact on modern optimi zation techniques has been the most pronounced. Chapters 5 and 6 are devoted to two characteristic aspects of duality theory: conjugate functions or polarity on the one hand, and saddle points on the other. The Farkas lemma on linear inequalities and its generalizations, Motzkin's description of polyhedra, Minkowski's supporting plane theorem are indispensable elementary tools which are contained in chapters 1, 2 and 3, respectively. The treatment of extremal properties of polyhedra as well as of general convex sets is based on the far reaching work of Klee. Chapter 2 terminates with a description of Gale diagrams, a recently developed successful technique for exploring polyhedral structures.

Inhalt
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 Fenchel's Duality Theorem.- 5.4. Proof of the Generalized Fenchel Theorem.- 5.5. Alternative Characterizations of Stability.- 5.6. Generation of Stable Functions.- 5.7. Rockafellar's 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.