# Linear and Integer Programming vs Linear Integration and Counting

## A Duality Viewpoint

• Jean-Bernard Lasserre
Book

1. Front Matter
Pages 1-11
2. Jean-Bemard Lasserre
Pages 1-5
3. ### Linear Integration and Linear Programming

1. Front Matter
Pages 7-7
2. Jean-Bemard Lasserre
Pages 9-29
3. Jean-Bemard Lasserre
Pages 31-37
4. ### Linear Counting and Integer Programming

1. Front Matter
Pages 39-39
2. Jean-Bemard Lasserre
Pages 41-69
3. Jean-Bemard Lasserre
Pages 71-79
5. ### Duality

1. Front Matter
Pages 81-81
2. Jean-Bemard Lasserre
Pages 83-106
3. Jean-Bemard Lasserre
Pages 107-113
4. Jean-Bemard Lasserre
Pages 115-129
5. Jean-Bemard Lasserre
Pages 131-138
6. Jean-Bemard Lasserre
Pages 139-147
6. Back Matter
Pages 1-18

### Introduction

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.

### Keywords

Brion Vergne algorithms interger programming linear integration linear optimization linear programming linear summation operations research optimization

#### Authors and affiliations

• Jean-Bernard Lasserre

There are no affiliations available

### Bibliographic information

• DOI https://doi.org/10.1007/978-0-387-09414-4
• Copyright Information Springer-Verlag New York 2009
• Publisher Name Springer, New York, NY
• eBook Packages Mathematics and Statistics
• Print ISBN 978-0-387-09413-7
• Online ISBN 978-0-387-09414-4
• Series Print ISSN 1431-8598