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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.
Established Tools of Discrete Optimization
Tools from Linear and Convex Optimization
Tools from the Geometry of Numbers and Integer Optimization
Graver Basis Methods
Graver Bases
Graver Bases for Block-Structured Integer Programs
Generating Function Methods
Introduction to Generating Functions
Decompositions of Indicator Functions of Polyhedral
Barvinok's Short Rational Generating Functions
Global Mixed-Integer Polynomial Optimization via Summation
Multicriteria Integer Linear Optimization via Integer Projection
Gröbner Basis Methods
Computations with Polynomials
Gröbner Bases in Integer Programming
Nullstellensatz and Positivstellensatz Relaxations
The Nullstellensatz in Discrete Optimization
Positivity of Polynomials and Global Optimization
Epilogue