This monograph develops an algorithmic theory of nonlinear discrete optimization. It introduces a simple and useful setup, which enables the polynomial time solution of broad fundamental classes of nonlinear combinatorial optimization and integer programming problems in variable dimension. An important part of this theory is enhanced by recent developments in the algebra of Graver bases. The power of the theory is demonstrated by deriving the first polynomial time algorithms in a variety of application areas within operations research and statistics, including vector partitioning, matroid optimization, experimental design, multicommodity flows, multi-index transportation and privacy in statistical databases. This monograph is intended for graduate students and researchers. It is accessible to anyone with standard undergraduate knowledge and mathematical maturity.

In recent years, many new techniques have emerged in the mathematical theory of discrete optimization that have proven to be effective in solving a number of hard problems. This book presents these recent advances, particularly those that arise from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside of the standard curriculum in optimization. These new techniques, all of which are presented with minimal prerequisites, provide a transition from linear to nonlinear discrete optimization. This book can be used as a textbook for advanced undergraduates or first-year graduate students in mathematics, computer science or operations research. It is also appropriate for mathematicians, engineers, and scientists engaged in computation who wish to gain a deeper understanding of how and why algorithms work.

Interest in constrained optimization originated with the simple linear pro gramming model since it was practical and perhaps the only computationally tractable model at the time. Constrained linear optimization models were soon adopted in numerous application areas and are perhaps the most widely used mathematical models in operations research and management science at the time of this writing. Modelers have, however, found the assumption of linearity to be overly restrictive in expressing the real-world phenomena and problems in economics, finance, business, communication, engineering design, computational biology, and other areas that frequently demand the use of nonlinear expressions and discrete variables in optimization models. Both of these extensions of the linear programming model are NP-hard, thus representing very challenging problems. On the brighter side, recent advances in algorithmic and computing technology make it possible to re visit these problems with the hope of solving practically relevant problems in reasonable amounts of computational time. Initial attempts at solving nonlinear programs concentrated on the de velopment of local optimization methods guaranteeing globality under the assumption of convexity. On the other hand, the integer programming liter ature has concentrated on the development of methods that ensure global optima. The aim of this book is to marry the advancements in solving nonlinear and integer programming models and to develop new results in the more general framework of mixed-integer nonlinear programs (MINLPs) with the goal of devising practically efficient global optimization algorithms for MINLPs.

This book aims to demonstrate and detail the pervasive nature of Discrete Optimization. The handbook couples the difficult, critical-thinking aspects of mathematical modeling with the hot area of discrete optimization. It is done with an academic treatment outlining the state-of-the-art for researchers across the domains of the Computer Science, Math Programming, Applied Mathematics, Engineering, and Operations Research. The book utilizes the tools of mathematical modeling, optimization, and integer programming to solve a broad range of modern problems.

Optimization is the act of obtaining the "best" result under given circumstances. In design, construction, and maintenance of any engineering system, engineers must make technological and managerial decisions to minimize either the effort or cost required or to maximize benefits. There is no single method available for solving all optimization problems efficiently. Several optimization methods have been developed for different types of problems. The optimum-seeking methods are mathematical programming techniques (specifically, nonlinear programming techniques). Nonlinear Optimization: Models and Applications presents the concepts in several ways to foster understanding. Geometric interpretation: is used to re-enforce the concepts and to foster understanding of the mathematical procedures. The student sees that many problems can be analyzed, and approximate solutions found before analytical solutions techniques are applied. Numerical approximations: early on, the student is exposed to numerical techniques. These numerical procedures are algorithmic and iterative. Worksheets are provided in Excel, MATLAB®, and MapleTM to facilitate the procedure. Algorithms: all algorithms are provided with a step-by-step format. Examples follow the summary to illustrate its use and application. Nonlinear Optimization: Models and Applications: Emphasizes process and interpretation throughout Presents a general classification of optimization problems Addresses situations that lead to models illustrating many types of optimization problems Emphasizes model formulations Addresses a special class of problems that can be solved using only elementary calculus Emphasizes model solution and model sensitivity analysis About the author: William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. He received his Ph.D. at Clemson University and has taught at the United States Military Academy and at Francis Marion University where he was the chair of mathematics. He has written many publications, including over 20 books and over 150 journal articles. Currently, he is an adjunct professor in the Department of Mathematics at the College of William and Mary. He is the emeritus director of both the High School Mathematical Contest in Modeling and the Mathematical Contest in Modeling.

This volume presents the fundamentals of nonlinear and mixed-integer optimisation, and their applications in the important area of process synthesis in chemical engineering. Topics that are unique include the theory and methods for mixed-integer nonlinear optimisation, introduction to modelling issues in process synthesis, and optimisation-based approaches in the synthesis of heat recovery systems, distillation-based systems, and reactor-based systems.

This is a supplementary volume to the major three-volume Handbook of Combinatorial Optimization set. It can also be regarded as a stand-alone volume presenting chapters dealing with various aspects of the subject in a self-contained way.

Network optimization is important in the modeling of problems and processes from such fields as engineering, computer science, operations research, transportation, telecommunication, decision support systems, manufacturing, and airline scheduling. Recent advances in data structures, computer technology, and algorithm development have made it possible to solve classes of network optimization problems that until recently were intractable. The refereed papers in this volume reflect the interdisciplinary efforts of a large group of scientists from academia and industry to model and solve complicated large-scale network optimization problems.

Many engineering, operations, and scientific applications include a mixture of discrete and continuous decision variables and nonlinear relationships involving the decision variables that have a pronounced effect on the set of feasible and optimal solutions. Mixed-integer nonlinear programming (MINLP) problems combine the numerical difficulties of handling nonlinear functions with the challenge of optimizing in the context of nonconvex functions and discrete variables. MINLP is one of the most flexible modeling paradigms available for optimization; but because its scope is so broad, in the most general cases it is hopelessly intractable. Nonetheless, an expanding body of researchers and practitioners — including chemical engineers, operations researchers, industrial engineers, mechanical engineers, economists, statisticians, computer scientists, operations managers, and mathematical programmers — are interested in solving large-scale MINLP instances.

This book provides the foundations of the theory of nonlinear optimization as well as some related algorithms and presents a variety of applications from diverse areas of applied sciences. The author combines three pillars of optimization?theoretical and algorithmic foundation, familiarity with various applications, and the ability to apply the theory and algorithms on actual problems?and rigorously and gradually builds the connection between theory, algorithms, applications, and implementation. Readers will find more than 170 theoretical, algorithmic, and numerical exercises that deepen and enhance the reader's understanding of the topics. The author includes offers several subjects not typically found in optimization books?for example, optimality conditions in sparsity-constrained optimization, hidden convexity, and total least squares. The book also offers a large number of applications discussed theoretically and algorithmically, such as circle fitting, Chebyshev center, the Fermat?Weber problem, denoising, clustering, total least squares, and orthogonal regression and theoretical and algorithmic topics demonstrated by the MATLAB? toolbox CVX and a package of m-files that is posted on the book?s web site.