Linear Programming Duality - An Introduction to Oriented Matroids

Der Begriff der Dualität in der linearen Programmierung ist von zentraler Bedeutung in der kombinatorischen Optimierung. Beide Autoren haben viele Jahre auf diesem Gebiet gearbeitet. Sie geben eine elementare Einführung in die Theorie der orientierten Matroide. Dabei gelingt es ihnen, die theoretischen Grundlagen der linearen Programmierung besonders klar herauszuarbeiten und die Beweise häufig verwendeter Resultate zu vereinfachen. Das Buch enthält viele Abbildungen, und die Autoren haben alle Kapitel mit Hinweisen auf weiterführende Literatur versehen.

Kurzbeschreibung:

This book presents an elementary introduction to the theory of oriented matroids. The way oriented matroids are introduced emphasizes that they are the most general - and hence simplest - structures for which Linear Programming Duality results can be stated and proved. The main theme of the book is duality. Using Farkas' Lemma as the basis the authors start with results on polyhedra in Rn and show how to restate the essence of the proofs in terms of sign patterns of oriented matroids. Most of the standard material in Linear Programming is presented in the setting of real space as well as in the more abstract theory of oriented matroids. This approach clarifies the theory behind Linear Programming and proofs become simpler. The last part of the book deals with the facial structure of polytopes respectively their oriented matroid counterparts. It is an introduction to more advanced topics in oriented matroid theory.
Each chapter contains suggestions for further reading and the references provide an overview of the research in this field.

Klappentext:

The main theorem of Linear Programming Duality, relating a "pri mal" Linear Programming problem to its "dual" and vice versa, can be seen as a statement about sign patterns of vectors in complemen tary subspaces of Rn. This observation, first made by R.T. Rockafellar in the late six ties, led to the introduction of certain systems of sign vectors, called "oriented matroids". Indeed, when oriented matroids came into being in the early seventies, one of the main issues was to study the fun damental principles underlying Linear Progra.mrning Duality in this abstract setting. In the present book we tried to follow this approach, i.e., rather than starting out from ordinary (unoriented) matroid theory, we pre ferred to develop oriented matroids directly as appropriate abstrac tions of linear subspaces. Thus, the way we introduce oriented ma troids makes clear that these structures are the most general -and hence, the most simple -ones in which Linear Programming Duality results can bestated and proved. We hope that this helps to get a better understanding of LP-Duality for those who have learned about it before und a good introduction for those who have not.