Optimization Over Integers

The book provides a unified, insightful, and modern treatment of the theory of integer optimization.

The book is used in the doctoral level course, "Integer and Combinatorial Optimization" at the Massachusetts Institute of Technology.

For solutions to exercises and other instructor resources, please contact Dimitris Bertsimas (dbertsim@mit.edu).

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 9-12 and develops cutting plane methods, integral basis methods, enumerative and heuristic 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 13 and 14, and treats mixed integer optimization and robust discrete optimization. Both areas are practically significant as real world problems have very often both continuous and discrete variables and have elements of uncertainty that need to be addressed in a tractable manner.

Distinguishing Characteristics Of This Book:

* Develops the theory of integer optimization from a new geometric perspective via integral generating sets;

* Emphasizes strong formulations, ways to improve them, integral polyhedra, duality, and relaxations;

* Discusses applications of lattices and algebraic geometry to integer optimization, including Grobner bases, optimization over polynomials and counting integer points in polyhedra;

* Contains a unified geometric treatment of cutting plane and integral basis methods;

* Covers enumerative and heuristic methods, including local search over exponential neighborhoods and simulated annealing;

* Presents the major methods to construct approximation algorithms: primal-dual, randomized rounding, semidefinite and enumerative methods;

* Provides a unified treatment of mixed integer and robust discrete optimization;

* Includes a large number of examples and exercises developed through extensive classroom use.