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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...

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Bibliographic Details
Main Authors: Stoer, Josef, Witzgall, Christoph (Author)
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1970, 1970
Edition:1st ed. 1970
Series:Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
Subjects:
Convex Geometry 
Convex And Discrete Geometry
Discrete Geometry
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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

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