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 optimizationn* Minimizes prerequisites for learning these methods. * Provides a transition from linear discrete optimization to nonlinear 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.
Jesus A. De Loera is a Professor of Mathematics and a member of the Graduate Groups in Computer Science and Applied Mathematics at University of California, Davis. His research has been recognised by an Alexander von Humboldt Fellowship, the UC Davis Chancellor Fellow award, and the 2010 INFORMS Computing Society Prize. He is an Associate Editor of SIAM Journal of Discrete Mathematics and Discrete Optimization. Raymond Hemmecke is a Professor of Combinatorial Optimization at Technische Universitat Munchen. His research interests include algebraic statistics, computer algebra and bioinformatics. Matthias Koppe is a Professor of mathematics and a member of the Graduate Groups in Computer Science and Applied Mathematics at University of California, Davis. He is an associate editor of Mathematical Programming, Series A and Asia-Pacific Journal of Operational Research.
* 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: Grobner Basis Methods* Chapter 10: Computations with Polynomials* Chapter 11: Grobner 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