Nonlinear Integer Programming

This book is a broad survey of theory and algorithms for the solution of nonlinear integer programming problems. These are optimization problems in which the objective function and constraint inequalities are nonlinear and the decision variables are restricted to integer values. The authors also consider mixed integer nonlinear programming problems in which some of the variables are continuous.

In the first five chapters, the authors introduce the theory of nonlinear integer programming, Lagrangian and surrogate duality for nonlinear integer programming problems, and the authors' own nonlinear Lagrangian Formulation. In the remaining chapters, the authors discuss methods for particular classes of nonlinear integer programming problems, including nonlinear knapsack problems, problems with quadratic objective functions, polynomial 0-1 programming problems, separable integer programming problems, nonseparable integer programming problems, and mixed integer nonlinear programming problems.

The main strength of this book is its broad coverage of methods for integer nonlinear nonlinear programming. The decision to organize the material into chapters on different classes of integer programming problems makes it easy for the reader to find an introduction to methods for a particular class of problems. The notes and bibliographic references that accompany each chapter are thorough and up to date. However, the book also has some significant weaknesses. The discussion of modeling and applications of nonlinear integer programming is limited. Mixed integer nonlinear programming is not adequately covered. The index is inadequate. Finally, there are many problems with the authors' English.

Compare this book with Floudas' Nonlinear and Mixed-Integer Optimization . Floudas' book provides better coverage of modeling, applications and algorithms for mixed integer nonlinear programming, while Li and Sun provide more thorough coverage of the research that has been done on pure integer nonlinear programming problems. The book by Li and Sun is recommended to readers who are specifically interested in methods for various classes of pure integer nonlinear programming problems.

References:

Christodoulos A. Floudas, Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, 1995.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.