Algebraic and Geometric Ideas in the Theory of Discrete Optimization

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.