Problems with multiple objectives and criteria are generally known as multiple criteria optimization or multiple criteria decision-making (MCDM) problems. So far, these types of problems have typically been modelled and solved by means of linear programming. However, many real-life phenomena are of a nonlinear nature, which is why we need tools for nonlinear programming capable of handling several conflicting or incommensurable objectives. In this case, methods of traditional single objective optimization and linear programming are not enough; we need new ways of thinking, new concepts, and new methods - nonlinear multiobjective optimization. Nonlinear Multiobjective Optimization provides an extensive, up-to-date, self-contained and consistent survey, review of the literature and of the state of the art on nonlinear (deterministic) multiobjective optimization, its methods, its theory and its background. The amount of literature on multiobjective optimization is immense. The treatment in this book is based on approximately 1500 publications in English printed mainly after the year 1980. Problems related to real-life applications often contain irregularities and nonsmoothnesses. The treatment of nondifferentiable multiobjective optimization in the literature is rather rare. For this reason, this book contains material about the possibilities, background, theory and methods of nondifferentiable multiobjective optimization as well. This book is intended for both researchers and students in the areas of (applied) mathematics, engineering, economics, operations research and management science; it is meant for both professionals and practitioners in many different fields of application. The intention has been to provide a consistent summary that may help in selecting an appropriate method for the problem to be solved. It is hoped the extensive bibliography will be of value to researchers.

Modeling, Control And Optimization Of Complex Systems is a collection of contributions from leading international researchers in the fields of dynamic systems, control theory, and modeling. These papers were presented at the Symposium on Modeling and Optimization of Complex Systems in honor of Larry Yu-Chi Ho in June 2001. They include exciting research topics such as:

-modeling of complex systems,
-power control in ad hoc wireless networks,
-adaptive control using multiple models,
-constrained control,
-linear quadratic control,
-discrete events,
-Markov decision processes and reinforcement learning,
-optimal control for discrete event and hybrid systems,
-optimal representation and visualization of multivariate data and functions in low-dimensional spaces.

The ASI on Nonlinear Model Based Process Control (August 10-20, 1997~ Antalya - Turkey) convened as a continuation of a previous ASI which was held in August 1994 in Antalya on Methods of Model Based Process Control in a more general context. In 1994, the contributions and discussions convincingly showed that industrial process control would increasingly rely on nonlinear model based control systems. Therefore, the idea for organizing this ASI was motivated by the success of the first one, the enthusiasm expressed by the scientific community for continuing contact, and the growing incentive for on-line control algorithms for nonlinear processes. This is due to tighter constraints and constantly changing performance objectives that now force the processes to be operated over a wider range of conditions compared to the past, and the fact that many of industrial operations are nonlinear in nature. The ASI intended to review in depth and in a global way the state-of-the-art in nonlinear model based control. The list of lecturers consisted of 12 eminent scientists leading the principal developments in the area, as well as industrial specialists experienced in the application of these techniques. Selected out of a large number of applications, there was a high quality, active audience composed of 59 students from 20 countries. Including family members accompanying the participants, the group formed a large body of92 persons. Out of the 71 participants, 11 were from industry.

Arc Routing: Theory, Solutions and Applications is about arc traversal and the wide variety of arc routing problems, which has had its foundations in the modern graph theory work of Leonhard Euler. Arc routing methods and computation has become a fundamental optimization concept in operations research and has numerous applications in transportation, telecommunications, manufacturing, the Internet, and many other areas of modern life. The book draws from a variety of sources including the traveling salesman problem (TSP) and graph theory, which are used and studied by operations research, engineers, computer scientists, and mathematicians. In the last ten years or so, there has been extensive coverage of arc routing problems in the research literature, especially from a graph theory perspective; however, the field has not had the benefit of a uniform, systematic treatment. With this book, there is now a single volume that focuses on state-of-the-art exposition of arc routing problems, that explores its graph theoretical foundations, and that presents a number of solution methodologies in a variety of application settings. Moshe Dror has succeeded in working with an elite group of ARC routing scholars to develop the highest quality treatment of the current state-of-the-art in arc routing.

The primary aim of this monograph is to provide a formal framework for the representation and management of uncertainty and vagueness in the field of artificial intelligence. It puts particular emphasis on a thorough analysis of these phenomena and on the development of sound mathematical modeling approaches. Beyond this theoretical basis the scope of the book includes also implementational aspects and a valuation of existing models and systems. The fundamental ambition of this book is to show that vagueness and un certainty can be handled adequately by using measure-theoretic methods. The presentation of applicable knowledge representation formalisms and reasoning algorithms substantiates the claim that efficiency requirements do not necessar ily require renunciation of an uncompromising mathematical modeling. These results are used to evaluate systems based on probabilistic methods as well as on non-standard concepts such as certainty factors, fuzzy sets or belief functions. The book is intended to be self-contained and addresses researchers and practioneers in the field of knowledge based systems. It is in particular suit able as a textbook for graduate-level students in AI, operations research and applied probability. A solid mathematical background is necessary for reading this book. Essential parts of the material have been the subject of courses given by the first author for students of computer science and mathematics held since 1984 at the University in Braunschweig."

One of the basic tenets of science is that deterministic systems are completely predictable-given the initial condition and the equations describing a system, the behavior of the system can be predicted 1 for all time. The discovery of chaotic systems has eliminated this viewpoint. Simply put, a chaotic system is a deterministic system that exhibits random behavior. Though identified as a robust phenomenon only twenty years ago, chaos has almost certainly been encountered by scientists and engi neers many times during the last century only to be dismissed as physical noise. Chaos is such a wide-spread phenomenon that it has now been reported in virtually every scientific discipline: astronomy, biology, biophysics, chemistry, engineering, geology, mathematics, medicine, meteorology, plasmas, physics, and even the social sci ences. It is no coincidence that during the same two decades in which chaos has grown into an independent field of research, computers have permeated society. It is, in fact, the wide availability of inex pensive computing power that has spurred much of the research in chaotic dynamics. The reason is simple: the computer can calculate a solution of a nonlinear system. This is no small feat. Unlike lin ear systems, where closed-form solutions can be written in terms of the system's eigenvalues and eigenvectors, few nonlinear systems and virtually no chaotic systems possess closed-form solutions."

There have been significant developments in the theory and practice of combinatorial optimization in the last 15 years. This progress has been evidenced by a continuously increasing number of international and local conferences, books and papers in this area. This book is also another contribution to this burgeoning area of operations research and optimization. This volume contains the contributions of the participants of the recent NATO Ad vanced Study Institute, New Frontiers in the Theory and Practice of Combinatorial Op timization, which was held at the campus of Bilkent University, in Ankara, Turkey, July 16-29, 1990. In this conference, we brought many prominent researchers and young and promising scientists together to discuss current and future trends in the theory and prac tice of combinatorial optimization. The Bilkent campus was an excellent environment for such an undertaking. Being outside of Ankara, the capital of Turkey, Bilkent University gave the participants a great opportunity for exchanging ideas and discussing new theories and applications without much distraction. One of the primary goals of NATO ASIs is to bring together a group of scientists and research scientists primarily from the NATO countries for the dissemination of ad vanced scientific knowledge and the promotion of international contacts among scientists. We believe that we accomplished this mission very successfully by bringing together 15 prominent lecturers and 45 promising young scientists from 12 countries, in a university environment for 14 days of intense lectures, presentations and discussions.

It is not an exaggeration to state that most problems dealt with in economic theory can be formulated as problems in optimization theory. This holds true for the paradigm of "behavioral" optimization in the pursuit of individual self interests and societally efficient resource allocation, as well as for equilibrium paradigms where existence and stability problems in dynamics can often be stated as "potential" problems in optimization. For this reason, books in mathematical economics and in mathematics for economists devote considerable attention to optimization theory. However, with very few exceptions, the reader who is interested in further study is left with the impression that there is no further place to go to and that what is in these second hand sources is all these is available as far as the subject of optimization theory is concerned. On the other hand the main results from mathematics are often carelessly stated or, more often than not, they do not get to be formally stated at all. Furthermore, it should be well understood that economic theory in general and, mathematical economics in particular, must be classified as special types of applied mathematics or, more precisely, of motivated mathematics since tools of mathematical analysis are used to prove theorems in an economics context in the manner in which probability theory may be classified. Hence, rigor and correct scholarship are of utmost importance and can not be subject to compromise.

During the academic year 2002-2003, the Faculty of Automatic Control and Computer Engineering of Ia~i (Romania), and its Departments of Automatic Control and Industrial Informatics and of Computer Engineering respectively, celebrated 25 years from the establishment of the specialization named Automatic Control and Computer Engineering within the framework of the former Faculty of Electrical Engineering of Ia~i, and, at the same time, 40 years since the first courses on Automatic Control and Computers respectively, were introduced in the curricula of the former specializations of Electromechanical Engineering and Electrical Power Engineering at the already mentioned Faculty of Electrical Engineering. The reader interested to know some important moments ofour evolution during the last five decades is invited to see the Addendum ofthis volume, where a short history is presented. And, to highlight once more the nice coincidences, it must be noted here that in 2003 our Technical University "Gheorghe Asachi" of Ia~i celebrated 190 years from the emergence of the first cadastral engineering degree course in Ia~i (thanks to the endeavor ofGheorghe Asachi), which is today considered to be the beginningofthe engineering higher education in Romania. Generally speaking, an anniversary is a celebration meant to mark special events ofthe past, with festivities to be performed solemnly and publicly according to a specific ritual.

This book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters."

This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. A variety of motivations leads mathematicians to study questions about attainment of the infimum in a minimization problem and its stability, duality and minmax theorems, convexification of sets and functions, and maximal monotone maps. For each there is the central problem of handling unbounded situations. Such problems arise in theory but also within the development of numerical methods. The book focuses on the notions of asymptotic cones and associated asymptotic functions that provide a natural and unifying framework for the resolution of these types of problems. These notions have been used largely and traditionally in convex analysis, yet these concepts play a prominent and independent role in both convex and nonconvex analysis. This book covers convex and nonconvex problems, offering detailed analysis and techniques that go beyond traditional approaches. The book will serve as a useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.

Complementarity theory is a new domain in applied mathematics and is concerned with the study of complementarity problems. These problems represent a wide class of mathematical models related to optimization, game theory, economic engineering, mechanics, fluid mechanics, stochastic optimal control etc. The book is dedicated to the study of nonlinear complementarity problems by topological methods. Audience: Mathematicians, engineers, economists, specialists working in operations research and anybody interested in applied mathematics or in mathematical modeling.

As our title reveals, we focus on optimal control methods and applications relevant to linear dynamic economic systems in discrete-time variables. We deal only with discrete cases simply because economic data are available in discrete forms, hence realistic economic policies should be established in discrete-time structures. Though many books have been written on optimal control in engineering, we see few on discrete-type optimal control. More over, since economic models take slightly different forms than do engineer ing ones, we need a comprehensive, self-contained treatment of linear optimal control applicable to discrete-time economic systems. The present work is intended to fill this need from the standpoint of contemporary macroeconomic stabilization. The work is organized as follows. In Chapter 1 we demonstrate instru ment instability in an economic stabilization problem and thereby establish the motivation for our departure into the optimal control world. Chapter 2 provides fundamental concepts and propositions for controlling linear deterministic discrete-time systems, together with some economic applica tions and numerical methods. Our optimal control rules are in the form of feedback from known state variables of the preceding period. When state variables are not observable or are accessible only with observation errors, we must obtain appropriate proxies for these variables, which are called "observers" in deterministic cases or "filters" in stochastic circumstances. In Chapters 3 and 4, respectively, Luenberger observers and Kalman filters are discussed, developed, and applied in various directions. Noticing that a separation principle lies between observer (or filter) and controller (cf."

The problem of obtaining dynamical models directly from an observed time-series occurs in many fields of application. There are a number of possible approaches to this problem. In this volume a number of such points of view are exposed: the statistical time series approach, a theory of guaranted performance, and finally a deterministic approximation approach. This volume is an out-growth of a number of get-togethers sponsered by the Systems and Decision Sciences group of the International Institute of Applied Systems Analysis (IIASA) in Laxenburg, Austria. The hospitality and support of this organization is gratefully acknowledged. Jan Willems Groningen, the Netherlands May 1989 TABLE OF CONTENTS Linear System Identification- A Survey page 1 M. Deistler A Tutorial on Hankel-Norm Approximation 26 K. Glover A Deterministic Approach to Approximate Modelling 49 C. Heij and J. C. Willems Identification - a Theory of Guaranteed Estimates 135 A. B. Kurzhanski Statistical Aspects of Model Selection 215 R. Shibata Index 241 Addresses of Authors 246 LINEAR SYSTEM IDENTIFICATION* A SURVEY M. DEISTLER Abstract In this paper we give an introductory survey on the theory of identification of (in general MIMO) linear systems from (discrete) time series data. The main parts are: Structure theory for linear systems, asymptotic properties of maximum likelihood type estimators, estimation of the dynamic specification by methods based on information criteria and finally, extensions and alternative approaches such as identification of unstable systems and errors-in-variables. Keywords Linear systems, parametrization, maximum likelihood estimation, information criteria, errors-in-variables.

This book is about time-domain modelling, stability, stabilization, control design and filtering for JTDS. It gives readers a thorough understanding of the basic mathematical analysis and fundamentals, offers a straightforward treatment of the different topics and provides broad coverage of the recent methodologies.

With contributions by specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics, the major themes include an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models.

The first part of a self-contained, elementary textbook, combining linear functional analysis, nonlinear functional analysis, numerical functional analysis, and their substantial applications with each other. As such, the book addresses undergraduate students and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems which relate to our real world. Applications concern ordinary and partial differential equations, the method of finite elements, integral equations, special functions, both the Schroedinger approach and the Feynman approach to quantum physics, and quantum statistics. As a prerequisite, readers should be familiar with some basic facts of calculus. The second part has been published under the title, Applied Functional Analysis: Main Principles and Their Applications.

Optimization in Quality Control presents a broad survey of the state of the art in optimization in quality, and focuses on industrial and national competitiveness. Each chapter has been carefully developed and refereed anonymously by experts in the area of optimization in quality control. Some of the topics covered in this volume include: * fundamentals of optimization techniques * contemporary approaches to optimization models in process control * economic design of control charts * determining optimal target values in multiple criteria economic selection models * examining quality improvement schemes by trading off between expected warranty servicing costs and increasing manufacturing costs * designing optimal inspection plans. This book will serve as an important reference source for academics, professionals and researchers.

My original introduction to this subject was through conservations, and ultimate ly joint work with C. A. Micchelli. I am grateful to him and to Profs. C. de Boor, E. W. Cheney, S. D. Fisher and A. A. Melkman who read various portions of the manuscript and whose suggestions were most helpful. Errors in accuracy and omissions are totally my responsibility. I would like to express my appreciation to the SERC of Great Britain and to the Department of Mathematics of the University of Lancaster for the year spent there during which large portions of the manuscript were written, and also to the European Research Office of the U.S. Army for its financial support of my research endeavors. Thanks are also due to Marion Marks who typed portions of the manuscript. Haifa, 1984 Allan Pinkus Table of Contents 1 Chapter I. Introduction . . . . . . . . Chapter II. Basic Properties of n-Widths . 9 1. Properties of d * * * * * * * * * * 9 n 15 2. Existence of Optimal Subspaces for d * n n 17 3. Properties of d * * * * * * 20 4. Properties of b * * * * * * n 5. Inequalities Between n-Widths 22 n 6. Duality Between d and d * * 27 n 7. n-Widths of Mappings of the Unit Ball 29 8. Some Relationships Between dn(T), dn(T) and bn(T) . 32 37 Notes and References . . . . . . . . . . . . . .

This handbook provides an in-depth examination of important theoretical methods and procedures in applied analysis. It details many of the most important theoretical trends in nonlinear analysis and applications to different fields. These features make the volume a valuable tool for every researcher working on nonlinear analysis.

Ever since the discovery of the five platonic solids in ancient times, the study of symmetry and regularity has been one of the most fascinating aspects of mathematics. Quite often the arithmetical regularity properties of an object imply its uniqueness and the existence of many symmetries. This interplay between regularity and symmetry properties of graphs is the theme of this book. Starting from very elementary regularity properties, the concept of a distance-regular graph arises naturally as a common setting for regular graphs which are extremal in one sense or another. Several other important regular combinatorial structures are then shown to be equivalent to special families of distance-regular graphs. Other subjects of more general interest, such as regularity and extremal properties in graphs, association schemes, representations of graphs in euclidean space, groups and geometries of Lie type, groups acting on graphs, and codes are covered independently. Many new results and proofs and more than 750 references increase the encyclopaedic value of this book.

This monograph deals with various classes of deterministic and stochastic continuous time optimal control problems that are defined over unbounded time intervals. For these problems the performance criterion is described by an improper integral and it is possible that, when evaluated at a given admissible element, this criterion is unbounded. To cope with this divergence new optimality concepts, referred to here as overtaking optimality, weakly overtaking optimality, agreeable plans, etc. , have been proposed. The motivation for studying these problems arises primarily from the economic and biological sciences where models of this type arise naturally. Indeed, any bound placed on the time hori zon is artificial when one considers the evolution of the state of an economy or species. The responsibility for the introduction of this interesting class of problems rests with the economists who first studied them in the modeling of capital accumulation processes. Perhaps the earliest of these was F. Ramsey [152] who, in his seminal work on the theory of saving in 1928, considered a dynamic optimization model defined on an infinite time horizon. Briefly, this problem can be described as a Lagrange problem with unbounded time interval. The advent of modern control theory, particularly the formulation of the famous Maximum Principle of Pontryagin, has had a considerable impact on the treat ment of these models as well as optimization theory in general.

In the late forties, Mathematical Programming became a scientific discipline in its own right. Since then it has experienced a tremendous growth. Beginning with economic and military applications, it is now among the most important fields of applied mathematics with extensive use in engineering, natural sciences, economics, and biological sciences. The lively activity in this area is demonstrated by the fact that as early as 1949 the first "Symposium on Mathe- matical Programming" took place in Chicago. Since then mathematical programmers from all over the world have gath- ered at the intfrnational symposia of the Mathematical Programming Society roughly every three years to present their recent research, to exchange ideas with their colleagues and to learn about the latest developments in their own and related fields. In 1982, the XI. International Symposium on Mathematical Programming was held at the University of Bonn, W. Germany, from August 23 to 27. It was organized by the Institut fUr Okonometrie und Operations Re- search of the University of Bonn in collaboration with the Sonderforschungs- bereich 21 of the Deutsche Forschungsgemeinschaft. This volume constitutes part of the outgrowth of this symposium and docu- ments its scientific activities. Part I of the book contains information about the symposium, welcoming addresses, lists of committees and sponsors and a brief review about the Ful- kerson Prize and the Dantzig Prize which were awarded during the opening ceremony.

Mathematics is playing an ever more important role in the physical and biologi cal sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modem as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and rein force the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematics Sci ences (AMS) series, which will focus on advanced textbooks and research-level monographs. v Preface This textbook introduces the basic concepts and results of mathematical control and system theory. Based on courses that I have taught during the last 15 years, it presents its subject in a self-contained and elementary fashion. It is geared primarily to an audience consisting of mathematically mature advanced undergraduate or beginning graduate students. In addi tion, it can be used by engineering students interested in a rigorous, proof oriented systems course that goes beyond the classical frequency-domain material and more applied courses."

Multilevel decision theory arises to resolve the contradiction between increasing requirements towards the process of design, synthesis, control and management of complex systems and the limitation of the power of technical, control, computer and other executive devices, which have to perform actions and to satisfy requirements in real time. This theory rises suggestions how to replace the centralised management of the system by hierarchical co-ordination of sub-processes. All sub-processes have lower dimensions, which support easier management and decision making. But the sub-processes are interconnected and they influence each other. Multilevel systems theory supports two main methodological tools: decomposition and co-ordination. Both have been developed, and implemented in practical applications concerning design, control and management of complex systems. In general, it is always beneficial to find the best or optimal solution in processes of system design, control and management. The real tendency towards the best (optimal) decision requires to present all activities in the form of a definition and then the solution of an appropriate optimization problem. Every optimization process needs the mathematical definition and solution of a well stated optimization problem. These problems belong to two classes: static optimization and dynamic optimization. Static optimization problems are solved applying methods of mathematical programming: conditional and unconditional optimization. Dynamic optimization problems are solved by methods of variation calculus: Euler Lagrange method; maximum principle; dynamical programming.