Nonlinear Integer Programming / International Series in Operations Research & Management Science Bd.84 (PDF)

From the reviews:

"This book addresses the topic of the general nonlinear integer programming (NLIP). The overall goal of the book is to bring the state of the art of the theoretical foundations and solution methods of NLIP to readers who are interested in optimization, operations research and computer science. This book investigates the theory and solution methodologies for the general NLIP and provides the developments of the last 30 years. It is assumed that readers are familiar with linear integer programming and the book thus focuses on the theory and solution methodologies of NLIP. The following are some of the features of the book: \roster \item"$\bullet$" Duality theory for NLIP: Investigation into the relationship of the duality gap and the perturbation function has lead to the development of nonlinear Lagrangian theory, thus establishing a methodology for the solution of the NLIP. \item"$\bullet$" Convergent Lagrangian and cutting plane methods for NLIP: Concepts like the objective level cuts, objective contour cuts or the domain cut reshapes the perturbation function. This leads to the optimal solution to lie in the convex hull of the perturbation functions and thus guarantees a zero duality gap. \item"$\bullet$" Convexification scheme: Using the connection between monotonicity and convexity, convexification schemes are developed for monotone and non-convex integer programs, thus extending the reach of branch and bound methods. \item"$\bullet$" Solution framework using global descent: The optimal solution of an NLIP is sought from among the local minima. A theoretical framework is also established to escape from a local minimum. \item"$\bullet$" Computational implications for NLIP: Several NLIPs with up to several thousand variables are solved by solution algorithms presented in this book."   (Romesh Saigal, Mathematical Reviews)