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Duan Li (Author), Xiaoling Sun (Author), "Nonlinear Integer Programming (International Series in Operations Research & Management Science)"

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.

- 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

"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) Complaints received. Download links deleted.

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