Revised throughout

Includes new chapters on the network simplex algorithm and a section on the five color theorem

Recent developments are discussed

Along with the traditional material concerning linear programming (the simplex method, the theory of duality, the dual simplex method), In-Depth Analysis of Linear Programming contains new results of research carried out by the authors. For the first time, the criteria of stability (in the geometrical and algebraic forms) of the general linear programming problem are formulated and proved. New regularization methods based on the idea of extension of an admissible set are proposed for solving unstable (ill-posed) linear programming problems. In contrast to the well-known regularization methods, in the methods proposed in this book the initial unstable problem is replaced by a new stable auxiliary problem. This is also a linear programming problem, which can be solved by standard finite methods. In addition, the authors indicate the conditions imposed on the parameters of the auxiliary problem which guarantee its stability, and this circumstance advantageously distinguishes the regularization methods proposed in this book from the existing methods. In these existing methods, the stability of the auxiliary problem is usually only presupposed but is not explicitly investigated. In this book, the traditional material contained in the first three chapters is expounded in much simpler terms than in the majority of books on linear programming, which makes it accessible to beginners as well as those more familiar with the area.

This book provides an introduction to the mathematical theory of optimization. It emphasizes the convergence theory of nonlinear optimization algorithms and applications of nonlinear optimization to combinatorial optimization. Mathematical Theory of Optimization includes recent developments in global convergence, the Powell conjecture, semidefinite programming, and relaxation techniques for designs of approximation solutions of combinatorial optimization problems.

The aim of the book is to cover the three fundamental aspects of research in equilibrium problems: the statement problem and its formulation using mainly variational methods, its theoretical solution by means of classical and new variational tools, the calculus of solutions and applications in concrete cases. The book shows how many equilibrium problems follow a general law (the so-called user equilibrium condition). Such law allows us to express the problem in terms of variational inequalities. Variational inequalities provide a powerful methodology, by which existence and calculation of the solution can be obtained.

Local search has been applied successfully to a diverse collection of optimization problems. However, results are scattered throughout the literature. This is the first book that presents a large collection of theoretical results in a consistent manner. It provides the reader with a coherent overview of the achievements obtained so far, and serves as a source of inspiration for the development of novel results in the challenging field of local search.

This book's aim is to provide several different kinds of information: a delineation of general metaheuristics methods, a number of state-of-the-art articles from a variety of well-known classical application areas as well as an outlook to modern computational methods in promising new areas. Therefore, this book may equally serve as a textbook in graduate courses for students, as a reference book for people interested in engineering or social sciences, and as a collection of new and promising avenues for researchers working in this field.

In the last decade several international conferences on Finsler, Lagrange and Hamilton geometries were organized in Bra ov, Romania (1994), Seattle, USA (1995), Edmonton, Canada (1998), besides the Seminars that periodically are held in Japan and Romania. All these meetings produced important progress in the field and brought forth the appearance of some reference volumes. Along this line, a new International Conference on Finsler and Lagrange Geometry took place August 26-31,2001 at the "Al.I.Cuza" University in Ia i, Romania. This Conference was organized in the framework of a Memorandum of Un derstanding (1994-2004) between the "Al.I.Cuza" University in Ia i, Romania and the University of Alberta in Edmonton, Canada. It was especially dedicated to Prof. Dr. Peter Louis Antonelli, the liaison officer in the Memorandum, an untired promoter of Finsler, Lagrange and Hamilton geometries, very close to the Romanian School of Geometry led by Prof. Dr. Radu Miron. The dedica tion wished to mark also the 60th birthday of Prof. Dr. Peter Louis Antonelli. With this occasion a Diploma was given to Professor Dr. Peter Louis Antonelli conferring the title of Honorary Professor granted to him by the Senate of the oldest Romanian University (140 years), the "Al.I.Cuza" University, Ia i, Roma nia. There were almost fifty participants from Egypt, Greece, Hungary, Japan, Romania, USA. There were scheduled 45 minutes lectures as well as short communications."

The volume, devoted to variational analysis and its applications, collects selected and refereed contributions, which provide an outline of the field. The meeting of the title "Equilibrium Problems and Variational Models," which was held in Erice (Sicily) in the period June 23 - July 2 2000, was the occasion of the presentation of some of these papers; other results are a consequence of a fruitful and constructive atmosphere created during the meeting. New results, which enlarge the field of application of variational analysis, are presented in the book; they deal with the vectorial analysis, time dependent variational analysis, exact penalization, high order deriva tives, geometric aspects, distance functions and log-quadratic proximal methodology. The new theoretical results allow one to improve in a remarkable way the study of significant problems arising from the applied sciences, as continuum model of transportation, unilateral problems, multicriteria spatial price models, network equilibrium problems and many others. As noted in the previous book "Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models," edited by F. Giannessi, A. Maugeri and P.M. Pardalos, Kluwer Academic Publishers, Vol. 58 (2001), the progress obtained by variational analysis has permitted to han dle problems whose equilibrium conditions are not obtained by the mini mization of a functional. These problems obey a more realistic equilibrium condition expressed by a generalized orthogonality (complementarity) con dition, which enriches our knowledge of the equilibrium behaviour. Also this volume presents important examples of this formulation."

Individuals and enterprises are looking for optimal solutions for the problems they face. Most problems can be expressed in mathematical terms, and so the methods of optimization render a significant aid. This book details the latest achievements in optimization. It offers comprehensive coverage on Differential Evolution, presenting revolutionary ideas in population-based optimization and shows the best known metaheuristics through the prism of Differential Evolution.

Proportional Optimization and Fairness is a long-needed attempt to reconcile optimization with apportionment in just-in-time (JIT) sequences and find the common ground in solving problems ranging from sequencing mixed-model just-in-time assembly lines through just-in-time batch production, balancing workloads in event graphs to bandwidth allocation internet gateways and resource allocation in computer operating systems. The book argues that apportionment theory and optimization based on deviation functions provide natural benchmarks for a process, and then looks at the recent research and developments in the field.

Individual chapters look at the theory of apportionment and just-in-time sequences; minimization of just-in-time sequence deviation; optimality of cyclic sequences and the oneness; bottleneck minimization; competition-free instances, Fraenkel's Conjecture, and optimal admission sequences; response time variability; applications to the Liu-Layland Problem and pinwheel scheduling; temporal capacity constraints and supply chain balancing; fair queuing and stride scheduling; and smoothing and batching.

This book concentrates on real-world production scheduling in factories and industrial settings. It includes industry case studies that use innovative techniques as well as academic research results that can be used to improve production scheduling. Its purpose is to present scheduling principles, advanced tools, and examples of innovative scheduling systems to persons who could use this information to improve their own production scheduling.

This text describes a series of models, propositions, and algorithms developed in recent years on time-varying networks. References and discussions on relevant problems and studies that have appeared in the literature are integrated in the book. Its eight chapters consider problems including the shortest path problem, the minimum-spanning tree problem, the maximum flow problem, and many more. The time-varying traveling salesman problem and the Chinese postman problem are presented in a chapter together with the time-varying generalized problem. While these topics are examined within the framework of time-varying networks, each chapter is self-contained so that each can be read - and used - separately.

Dynamic Programming in Economics is an outgrowth of a course intended for students in the first year PhD program and for researchers in Macroeconomics Dynamics. It can be used by students and researchers in Mathematics as well as in Economics. The purpose of Dynamic Programming in Economics is twofold: (a) to provide a rigorous, but not too complicated, treatment of optimal growth models in infinite discrete time horizon, (b) to train the reader to the use of optimal growth models and hence to help him to go further in his research. We are convinced that there is a place for a book which stays somewhere between the "minimum tool kit" and specialized monographs leading to the frontiers of research on optimal growth.

Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integration of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc.

Two short biographical articles on the works of Sobolev in the 1930s and the foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Sobolev are included.

This book develops models, results and algorithms for optimizing public transportation from a customer-oriented viewpoint. The methods used are based on graph-theoretic approaches and integer programming. The specific topics are all motivated by real-world examples which occurred in practical projects: location of stops, management of delay, and tariff zone design. An appendix summarizes some of the basics of optimization needed to interpret the material in the book.

Victor Isakov This volume contains various results on partial di?erential equations where Sobolev spaces are used. Their selection is motivated by the research int- ests of the editor and the geographicallinks to the places where S. L. Sobolev worked and lived: St. Petersburg, Moscow, and Novosibirsk. Most of the papers are written by leading experts in control theory and inverse pr- lems. Another reason for the selection is a strong link to applied areas. In my opinion, control theory and inverse problems are main areas of di?er- tial equations of importance for some branches of contemporary science and engineering. S. L. Sobolev, as many great mathematicians, was very much motivated by applications. He did not distinguished between pure and - plied mathematics, but, in his own words, between "good mathematics and bad mathematics. " While he possessed a brilliant analytical technique, he most valued innovative ideas, solutions of deep conceptual problems, and not mathematical decorations, perfecting exposition, and "generalizations. " S. L. Sobolev himself never published papers on inverse problems or c- trol theory, but he was very much aware of the state of art and he monitored research on inverse problems. In particular, in his lecture at a Conference on Di?erentialEquationsin1954(found inSobolev'sarchiveandmadeavailable to me by Alexander Bukhgeim), he outlined main inverse problems in g- physics: theinverseseismicproblem, theelectromagneticprospecting, andthe inverse problem of gravimetry.

This book provides successful implementations of metaheuristic methods for neural network training. It is the first book to achieve this objective. Moreover, the basic principles and fundamental ideas given in the book will allow the readers to create successful training methods on their own. Overall, the book's aim is to provide a broad coverage of the concepts, methods, and tools of the important area of ANNs within the realm of continuous optimization.

The primary objective underlying the Handbook on Modelling for Discrete Optimization is to demonstrate and detail the pervasive nature of Discrete Optimization. While its applications cut across an incredibly wide range of activities, many of the applications are only known to specialists. It is the aim of this handbook to correct this.

It has long been recognized that "modelling" is a critically important mathematical activity in designing algorithms for solving these discrete optimization problems. Nevertheless solving the resultant models is also often far from straightforward. In recent years it has become possible to solve many large-scale discrete optimization problems. However, some problems remain a challenge, even though advances in mathematical methods, hardware, and software technology have pushed the frontiers forward. This handbook couples the difficult, critical-thinking aspects of mathematical modelling with the hot area of discrete optimization. It will be done in an academic handbook treatment outlining the state-of-the-art for researchers across the domains of the Computer Science, Math Programming, Applied Mathematics, Engineering, and Operations Research. Included in the handbook's treatment are results from Graph Theory, Logic, Computer Science, and Combinatorics.

The chapters of this book are divided into two parts: (1) one dealing with general methods in the modelling of discrete optimization problems and (2) the other with specific applications. The first chapter of this volume, written by H. Paul Williams, can be regarded as a basic introduction of how to model discrete optimization problems as mixed integer problems, and outlines the main methods of solving them. In the second part of the book various real life applications are presented, most of them formulated as mixed integer linear or nonlinear programming problems. These applications include network problems, constant logic problems, many engineering problems, computer design, finance problems, medical diagnosis and medical treatment problems, applications of the Genome project, an array of transportation scheduling problems, and other applications.

Further information including a detailed Table of Contents and Preface can be found and examined on the Handbook's web page at http: //www.springer.com/0-387-32941-2.

Written by a world leader in the field and aimed at researchers in applied and engineering sciences, this brilliant text has as its main goal imparting an understanding of the methods so that practitioners can make immediate use of existing algorithms and software, and so that researchers can extend the state of the art and find new applications. It includes algorithms on seeking feasibility and analyzing infeasibility, as well as describing new and surprising applications.

This volume presents a wide range of medical applications that can utilize mathematical computing. This work grew out of a workshop on optimization which was held during the 2005 CIM Thematic Term on Optimization in Coimbra, Portugal. It provides an overview of the state-of-the-art in optimization in medicine and will serve as an excellent reference for researchers in the medical computing community and for those working in applied mathematics and optimization.

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.

This monograph is devoted to recent progress in the turnpike t- ory. Turnpike properties are well known in mathematical economics. The term was ?rst coined by Samuelson who showed that an e?cient expanding economy would for most of the time be in the vicinity of a balanced equilibrium path (also called a von Neumann path) [78, 79]. These properties were studied by many authors for optimal trajec- ries of a Neumann-Gale model determined by a superlinear set-valued mapping. In the monograph we discuss a number of results conce- ing turnpike properties in the calculus of variations and optimal control which were obtained by the author in the last ten years. These results showthattheturnpikepropertiesareageneralphenomenonwhichholds for various classes of variational problems and optimal control problems. Turnpike properties are studied for optimal control problems on- nite time intervals [T ,T ] of the real line. Solutions of such problems 1 2 (trajectories) always depend on the time interval [T ,T ], an optimality 1 2 criterion which is usually determined by a cost function, and on data which is some initial conditions. In the turnpike theory we are int- ested in the structure of solutions of optimal problems. We study the behavior of solutions when an optimality criterion is ?xed while T ,T 1 2 andthedatavary.

This is a comprehensive overview of the basics of fuzzy control, which also brings together some recent research results in soft computing, in particular fuzzy logic using genetic algorithms and neural networks.

This book offers researchers not only a solid background but also a snapshot of the current state of the art in this field.

This volume consists of papers presented in the special sessions on "Complex and Numerical Analysis," "Value Distribution Theory and Complex Domains," and "Use of Symbolic Computation in Mathematics Education" of the ISAAC'97 Congress held at the University of Delaware, during June 2-7, 1997. The ISAAC Congress coincided with a U.S.-Japan Seminar also held at the University of Delaware. The latter was supported by the National Science Foundation through Grant INT-9603029 and the Japan Society for the Promotion of Science through Grant MTCS-134. It was natural that the participants of both meetings should interact and consequently several persons attending the Congress also presented papers in the Seminar. The success of the ISAAC Congress and the U.S.-Japan Seminar has led to the ISAAC'99 Congress being held in Fukuoka, Japan during August 1999. Many of the same participants will return to this Seminar. Indeed, it appears that the spirit of the U.S.-Japan Seminar will be continued every second year as part of the ISAAC Congresses. We decided to include with the papers presented in the ISAAC Congress and the U.S.-Japan Seminar several very good papers by colleagues from the former Soviet Union. These participants in the ISAAC Congress attended at their own expense.

In the fifties and sixties, several real problems, old and new, especially in Physics, Mechanics, Fluidodynamics, Structural Engi- neering, have shown the need of new mathematical models for study- ing the equilibrium of a system. This has led to the formulation of Variational Inequalities (by G. Stampacchia), and to the develop- ment of Complementarity Systems (by W.S. Dorn, G.B. Dantzig, R.W. Cottle, O.L. Mangasarian et al.) with important applications in the elasto-plastic field (initiated by G. Maier). The great advan- tage of these models is that the equilibrium is not necessarily the extremum of functional, like energy, so that no such functional must be supposed to exist. In the same decades, in some fields like Control Theory, Net- works, Industrial Systems, Logistics, Management Science, there has been a strong request of mathmatical models for optimizing situa- tions where there are concurrent objectives, so that Vector Optimiza- tion (initiated by W. Pareto) has received new impetus. With regard to equilibrium problems, Vector Optimization has the above - mentioned drawback of being obliged to assume the exis- tence of a (vector) functional. Therefore, at the end of the seventies the study of Vector Variational Inequalities began with the scope of exploiting the advantages of both variational and vector models. This volume puts together most of the recent mathematical results in Vector Variational Inequalities with the purpose of contributing to further research.