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Algebraic and Geometric Ideas in the Theory of Discrete Optimization
TLDR
Algebraic and Geometric Ideas in the Theory of Discrete Optimization offers several research technologies not yet well known among practitioners of discrete optimization, minimizes prerequisites for learning these methods, and provides a transition from linear discrete optimization to nonlinear discrete optimization.Abstract:
This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside the standard curriculum in optimization. Algebraic and Geometric Ideas in the Theory of Discrete Optimization offers several research technologies not yet well known among practitioners of discrete optimization, minimizes prerequisites for learning these methods, and provides a transition from linear discrete optimization to nonlinear discrete optimization. Audience: This book can be used as a textbook for advanced undergraduates or beginning graduate students in mathematics, computer science, or operations research or as a tutorial for mathematicians, engineers, and scientists engaged in computation who wish to delve more deeply into how and why algorithms do or do not work. Contents: Part I: Established Tools of Discrete Optimization; Chapter 1: Tools from Linear and Convex Optimization; Chapter 2: Tools from the Geometry of Numbers and Integer Optimization; Part II: Graver Basis Methods; Chapter 3: Graver Bases; Chapter 4: Graver Bases for Block-Structured Integer Programs; Part III: Generating Function Methods; Chapter 5: Introduction to Generating Functions; Chapter 6: Decompositions of Indicator Functions of Polyhedral; Chapter 7: Barvinok s Short Rational Generating Functions; Chapter 8: Global Mixed-Integer Polynomial Optimization via Summation; Chapter 9: Multicriteria Integer Linear Optimization via Integer Projection; Part IV: Grbner Basis Methods; Chapter 10: Computations with Polynomials; Chapter 11: Grbner Bases in Integer Programming; Part V: Nullstellensatz and Positivstellensatz Relaxations; Chapter 12: The Nullstellensatz in Discrete Optimization; Chapter 13: Positivity of Polynomials and Global Optimization; Chapter 14: Epilogue.read more
Citations
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Sparse Signal Processing Concepts for Efficient 5G System Design
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Posted Content
An Algorithmic Theory of Integer Programming
Friedrich Eisenbrand,Christoph Hunkenschröder,Kim-Manuel Klein,Martin Koutecký,Asaf Levin,Shmuel Onn +5 more
TL;DR: It is shown that integer programming can be solved in time, and a strongly-polynomial algorithm is derived, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data.
Journal ArticleDOI
Scheduling meets n-fold Integer Programming
Dušan Knop,Martin Koutecký +1 more
TL;DR: In this article, it was shown that n-fold integer programming (NIFP) is fixed parameter tractable for some natural parameters, such as the maximum processing time of a job and the maximum weight of the job.
Reference BookDOI
Monomial Algebras, Second Edition
TL;DR: In this paper, the authors show how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs, and directly link the algebraic properties of monomial algebraic structures to combinatorical structures (such as simplicial complexes, posets, digraphs, graphs, and clutters).