Jesús 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 Universität München. His research interests include algebraic statistics, computer algebra and bioinformatics. Matthias Köppe 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 OptimizationChapter 1: Tools from Linear and Convex OptimizationChapter 2: Tools from the Geometry of Numbers and Integer OptimizationPart II: Graver Basis MethodsChapter 3: Graver BasesChapter 4: Graver Bases for Block-Structured Integer ProgramsPart III: Generating Function MethodsChapter 5: Introduction to Generating FunctionsChapter 6: Decompositions of Indicator Functions of PolyhedralChapter 7: Barvinok’s Short Rational Generating FunctionsChapter 8: Global Mixed-Integer Polynomial Optimization via SummationChapter 9: Multicriteria Integer Linear Optimization via Integer ProjectionPart IV: Gröbner Basis MethodsChapter 10: Computations with PolynomialsChapter 11: Gröbner Bases in Integer ProgrammingPart V: Nullstellensatz and Positivstellensatz RelaxationsChapter 12: The Nullstellensatz in Discrete OptimizationChapter 13: Positivity of Polynomials and Global OptimizationChapter 14: Epilogue