Jean-Bernard Lasserre (auth.)9780387094137, 038709413X
In this book the author analyzes and compares four closely related problems, namely linear programming, integer programming, linear integration, linear summation (or counting). The focus is on duality and the approach is rather novel as it puts integer programming in perspective with three associated problems, and permits one to define discrete analogues of well-known continuous duality concepts, and the rationale behind them. Also, the approach highlights the difference between the discrete and continuous cases. Central in the analysis are the continuous and discrete Brion and Vergne’s formulae for linear integration and counting. This approach provides some new insights on duality concepts for integer programs, and also permits to retrieve and shed new light on some well-known results. For instance, Gomory relaxations and the abstract superadditive dual of integer programs are re-interpreted in this algebraic approach.
This book will serve graduate students and researchers in applied mathematics, optimization, operations research and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will also find this book useful.
Table of contents :
Front Matter….Pages 1-11
Introduction….Pages 1-5
Front Matter….Pages 7-7
The Linear Integration Problem I….Pages 9-29
Comparing the Continuous Problems P and I….Pages 31-37
Front Matter….Pages 39-39
The Linear Counting Problem I d ….Pages 41-69
Relating the Discrete Problems P d and I d with P….Pages 71-79
Front Matter….Pages 81-81
Duality and Gomory Relaxations….Pages 83-106
Barvinok’s Counting Algorithm and Gomory Relaxations….Pages 107-113
A Discrete Farkas Lemma….Pages 115-129
The Integer Hull of a Convex Rational Polytope….Pages 131-138
Duality and Superadditive Functions….Pages 139-147
Back Matter….Pages 1-18
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