Synopses & Reviews
The methodological development of integer programming has grown by leaps and bounds in the past four decades, with its main focus on linear integer programming. However, the past few years have also witnessed certain promising theoretical and methodological achievements in nonlinear integer programming. These recent developments have produced applications of nonlinear (mixed) integer programming across a variety of various areas of scientific computing, engineering, management science and operations research. Its prominent applications include, for examples, portfolio selection, capital budgeting, production planning, resource allocation, computer networks, reliability networks and chemical engineering. In recognition of nonlinearity's academic significance in optimization and its importance in real world applications, NONLINEAR INTEGER PROGRAMMING is a comprehensive and systematic treatment of the methodology. The book's goal is to bring the state-of-the-art of the theoretical foundation and solution methods for nonlinear integer programming to students and researchers in optimization, operations research, and computer science. This book systemically investigates theory and solution methodologies for general nonlinear integer programming, and at the same time, provides a timely and comprehensive summary of the theoretical and algorithmic development in the last 30 years on this topic. The following are some features of the book: Duality theory for nonlinear integer programming is thoroughly discussed. Convergent Lagrangian and cutting methods for separable nonlinear integer programming are explained and demonstrated. Convexification scheme and the relation between the monotonicity and convexity is explored and illustrated. A solution framework is provided using global descent. Computational implementations for large-scale nonlinear integer programming problems are demonstrated for several efficient solution algorithms presented in the book.
Review
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) "The book's goal is to bring the state-of-the-art NIP theoretical foundation and solution methods to students and researchers in optimization, OR, and computer science. ... I think this is an interesting book about NIP and related issues. Many students, professionals, professors, and researchers could use it to learn key concepts about this topic. Therefore, I consider this work a valuable reference for graduate students or researchers interested in integer programming in a wide sense ... ." (Javier Faulin, Interfaces, Vol. 37 (4), 2007) "The goal of the book is to bring the state-of-the-art of the theoretical foundation and solution methods for nonlinear integer programming to readers who are interested in optimization, operations research and computer science. ... Readers of this book can be researchers, practitioners, graduate students and senior undergraduate students ... . It can be also used as a reference book for researchers, engineers and practitioners to solve real-world application problems by nonlinear integer programming models." (Oleg A. Shcherbina, Zentralblatt MATH, Vol. 1140, 2008)
Synopsis
It is not an exaggeration that much of what people devote in their hfe re solves around optimization in one way or another. On one hand, many decision making problems in real applications naturally result in optimization problems in a form of integer programming. On the other hand, integer programming has been one of the great challenges for the optimization research community for many years, due to its computational difficulties: Exponential growth in its computational complexity with respect to the problem dimension. Since the pioneering work of R. Gomory 80] in the late 1950s, the theoretical and methodological development of integer programming has grown by leaps and bounds, mainly focusing on linear integer programming. The past few years have also witnessed certain promising theoretical and methodological achieve ments in nonlinear integer programming. When the first author of this book was working on duality theory for n- convex continuous optimization in the middle of 1990s, Prof. Douglas J. White suggested that he explore an extension of his research results to integer pro gramming. The two authors of the book started their collaborative work on integer programming and global optimization in 1997. The more they have investigated in nonlinear integer programming, the more they need to further delve into the subject. Both authors have been greatly enjoying working in this exciting and challenging field."
Synopsis
A combination of both Integer Programming and Nonlinear Optimization, this is a powerful book that surveys the field and provides a state-of-the-art treatment of Nonlinear Integer Programming. It is the first book available on the subject. The book aims to bring the theoretical foundation and solution methods for nonlinear integer programming to students and researchers in optimization, operations research, and computer science.
Table of Contents
List of figures.- List of tables.- Preface.- Acknowledgments.- Introduction.- Optimality, relaxation and general solution procedures.- Lagrangian duality theory.- Surrogate duality theory.- Nonlinear Lagrangian and strong duality.- Nonlinear knapsack problems.- Separable integer programming.- Nonlinear integer programming with a quadratic objective function.- Nonseparable integer programming.- Unconstrained polynomial 0-1 optimization.- Constrained polynomial 0-1 programming.- Two level methods for constrained polynomial 0-1 programming.- Mixed-integer nonlinear programming.- Global descent methods.- References.- Index.