Linear and Integer Programming vs Linear Integration and Counting [electronic resource] : A Duality Viewpoint / by Jean-Bernard Lasserre.

By: Lasserre, Jean-Bernard [author.]
Contributor(s): SpringerLink (Online service)
Material type: TextTextSeries: Springer Series in Operations Research and Financial Engineering: Publisher: New York, NY : Springer New York, 2009Description: XIV, 168 p. 2 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780387094144Subject(s): Mathematics | Computer science -- Mathematics | Convex geometry | Discrete geometry | Mathematical optimization | Operations research | Management science | Mathematics | Operations Research, Management Science | Optimization | Discrete Mathematics in Computer Science | Convex and Discrete GeometryAdditional physical formats: Printed edition:: No titleDDC classification: 519.6 LOC classification: QA402-402.37T57.6-57.97Online resources: Click here to access online
Contents:
I Linear Integration and Linear Programming -- The Linear Integration Problem I -- Comparing the Continuous Problems P and I -- II Linear Counting and Integer Programming -- The Linear Counting Problem I -- Relating the Discrete Problems P and I with P -- III Duality -- Duality and Gomory Relaxations -- Barvinok#x2019;s Counting Algorithm and Gomory Relaxations -- A Discrete Farkas Lemma -- The Integer Hull of a Convex Rational Polytope -- Duality and Superadditive Functions.
In: Springer eBooksSummary: 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.
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I Linear Integration and Linear Programming -- The Linear Integration Problem I -- Comparing the Continuous Problems P and I -- II Linear Counting and Integer Programming -- The Linear Counting Problem I -- Relating the Discrete Problems P and I with P -- III Duality -- Duality and Gomory Relaxations -- Barvinok#x2019;s Counting Algorithm and Gomory Relaxations -- A Discrete Farkas Lemma -- The Integer Hull of a Convex Rational Polytope -- Duality and Superadditive Functions.

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.

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