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Trends in Optimization

American Mathematical Society Short Course, January 5-6, 2004, Phoeniz, Arizona

By: American Mathematical Society. Short Course,Serkan Hoşten,Jon Lee,Rekha R. Thomas,American Mathematical Society

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Trends in Optimization

American Mathematical Society Short Course, January 5-6, 2004, Phoeniz, Arizona

By: American Mathematical Society. Short Course,Serkan Hoşten,Jon Lee,Rekha R. Thomas,American Mathematical Society

This volume presents proceedings from the AMS short course, Trends in Optimization 2004, held at the Joint Mathematics Meetings in Phoenix (AZ). It focuses on seven exciting areas of discrete optimization. In particular, Karen Aardal describes Lovasz's fundamental algorithm for producing a short vector in a lattice by basis reduction and H.W. Lenstra's use of this idea in the early 1980s in his polynomial-time algorithm for integer programming in fixed dimension. Aardal's article, lucid presentations of the material. It also contains practical developments using computational tools. Bernd Sturmfels' article, Algebraic recipes for integer programming, discusses how methods of commutative algebra and algebraic combinatorics can be used successfully to attack integer programming problems. Specifically, Grobner bases play a central role in algorithmic theory and practice. Moreover, it is shown that techniques based on short rational functions are bringing new insights, such as in computing the integer programming gap. Overall, these articles, together with five other contributions, make this volume an impressive compilation on the state-of-the-art of optimization. It is suitable for graduate students and researchers interested in discrete optimization.

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Page Count

154

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082183584X

ISBN 13

9780821835845

Lectures in Geometric Combinatorics

By: Rekha R. Thomas

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Lectures in Geometric Combinatorics

By: Rekha R. Thomas

This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the statepolytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Grobner bases of toric ideals and other methods fromcommutative algebra. The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes.

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Page Count

156

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821841408

ISBN 13

9780821841402