Convexity and Optimization in Finite Dimensions I

1. Front Matter

   Pages I-IX

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2. Introduction

   • Josef Stoer, Christoph Witzgall

   Pages 1-6

3. Inequality Systems

   • Josef Stoer, Christoph Witzgall

   Pages 7-30

4. Convex Polyhedra

   • Josef Stoer, Christoph Witzgall

   Pages 31-81

5. Convex Sets

   • Josef Stoer, Christoph Witzgall

   Pages 82-133

6. Convex Functions

   • Josef Stoer, Christoph Witzgall

   Pages 134-176

7. Duality Theorems

   • Josef Stoer, Christoph Witzgall

   Pages 177-220

8. Saddle Point Theorems

   • Josef Stoer, Christoph Witzgall

   Pages 221-268

9. Back Matter

   Pages 269-298

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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.

• Arithmetic
• Convexity
• Dimensions
• Finite
• Konvexe Planungsrechnung
• Topology
• algebra
• function
• geometry
• theorem

• Universität Würzburg, Deutschland

  Josef Stoer

• Boeing Scientific Research Laboratories, Seattle, USA

  Christoph Witzgall

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