Co-author: Robert Weismantel
Dynamic Ideas, Belmont, Massachusetts, January, 2005.
The purpose of this book is to provide a unified, insightful, and modern treatment of the theory of integer optimization with an eye towards the
future. We have selected those topics that we feel have influenced the current state of the art and most importantly we feel will affect the future of the field. We depart from earlier treatments of integer optimization by placing significant emphasis on strong formulations, duality, algebra and most importantly geometry.
The chapters of the book are logically organized in four parts:
Part I: Formulations and relaxations includes Chapters 1-5 and discusses how to formulate integer optimization problems, how to enhance the formulations to improve the quality of relaxations, how to obtain ideal formulations, the duality of integer optimization and how to solve the resulting relaxations both practically and theoretically.
Part II: Algebra and geometry of integer optimization includes Chapters 6-8 and
develops the theory of lattices, oulines ideas from algebraic geometry that have had an impact on integer optimization, and most importantly discusses the geometry of integer optimization, a key feature of the book. These chapters provide the building blocks for developing algorithms.
Part III: Algorithms for integer optimization includes Chapters 9-11 and develops
cutting plane methods, integral basis methods, enumerative methods and approximation
algorithms. The key characteristic of our treatment is that our development of the algorithms is naturally based on the algebraic and geometric developments of Part II.
Part IV: Extensions of integer optimization includes Chapters 12 and 13, and treats
mixed integer optimization and robust discrete optimization. Both areas are practically significant as real world problems have very often both continous and discrete variables and have elements of uncertainty that need to be addressed in a tractable manner.