This volume of research papers comprises the proceedings of the first International Conference on Mathematics of Neural Networks and Applications (MANNA), which was held at Lady Margaret Hall, Oxford from July 3rd to 7th, 1995 and attended by 116 people. The meeting was strongly supported and, in addition to a stimulating academic programme, it featured a delightful venue, excellent food and accommo dation, a full social programme and fine weather - all of which made for a very enjoyable week. This was the first meeting with this title and it was run under the auspices of the Universities of Huddersfield and Brighton, with sponsorship from the US Air Force (European Office of Aerospace Research and Development) and the London Math ematical Society. This enabled a very interesting and wide-ranging conference pro gramme to be offered. We sincerely thank all these organisations, USAF-EOARD, LMS, and Universities of Huddersfield and Brighton for their invaluable support. The conference organisers were John Mason (Huddersfield) and Steve Ellacott (Brighton), supported by a programme committee consisting of Nigel Allinson (UMIST), Norman Biggs (London School of Economics), Chris Bishop (Aston), David Lowe (Aston), Patrick Parks (Oxford), John Taylor (King's College, Lon don) and Kevin Warwick (Reading). The local organiser from Huddersfield was Ros Hawkins, who took responsibility for much of the administration with great efficiency and energy. The Lady Margaret Hall organisation was led by their bursar, Jeanette Griffiths, who ensured that the week was very smoothly run."

Linear Programming (LP) is perhaps the most frequently used optimization technique. One of the reasons for its wide use is that very powerful solution algorithms exist for linear optimization. Computer programs based on either the simplex or interior point methods are capable of solving very large-scale problems with high reliability and within reasonable time. Model builders are aware of this and often try to formulate real-life problems within this framework to ensure they can be solved efficiently. It is also true that many real-life optimization problems can be formulated as truly linear models and also many others can well be approximated by linearization. The two main methods for solving LP problems are the variants of the simplex method and the interior point methods (IPMs). It turns out that both variants have their role in solving different problems. It has been recognized that, since the introduction of the IPMs, the efficiency of simplex based solvers has increased by two orders of magnitude. This increased efficiency can be attributed to the following: (1) theoretical developments in the underlying algorithms, (2) inclusion of results of computer science, (3) using the principles of software engineering, and (4) taking into account the state-of-the-art in computer technology.
Theoretically correct algorithms can be implemented in many different ways, but the performance is dependent on how the implementation is done. The success is based on the proper synthesis of the above mentioned (1-4) components. Computational Techniques of the Simplex Method is a systematic treatment focused on the computational issues of the simplex method. It provides a comprehensive coverage of the most important and successful algorithmic and implementation techniques of the simplex method. It is a unique source of essential, never discussed details of algorithmic elements and their implementation. On the basis of the book the reader will be able to create a highly advanced implementation of the simplex method which, in turn, can be used directly or as a building block in other solution algorithms.

This book arose out of an invited feature article on visualization and opti mization that appeared in the ORSA Journal on Computing in 1994. That article briefly surveyed the current state of the art in visualization as it ap plied to optimization. In writing the feature article, it became clear that there was much more to say. Apparently others agreed, and thus this book was born. The book is targeted primarily towards the optimization community rather than the visualization community. Although both optimization and visualization both seek to help people understand complex problems, prac titioners in one field are generally unaware of work in the other field. Given the common goals of the respective fields, it seemed fruitful to consider how each can contribute to the other. One might argue that this book should not be focused specifically on optimization but on decision making in general. Perhaps, but it seems that there is sufficient material to create a book targeted specifically to optimization. Certainly many of the ideas presented in the book are appli cable to other areas, including computer simulation, decision theory and stochastic modeling. Another book could discuss the use of visualization in these areas.

The book Scatter Search by Manuel Laguna and Rafael Marti represents a long-awaited "missing link" in the literature of evolutionary methods. Scatter Search (SS)-together with its generalized form called Path Relinking-constitutes the only evolutionary approach that embraces a collection of principles from Tabu Search (TS), an approach popularly regarded to be divorced from evolutionary procedures. The TS perspective, which is responsible for introducing adaptive memory strategies into the metaheuristic literature (at purposeful level beyond simple inheritance mechanisms), may at first seem to be at odds with population-based approaches. Yet this perspective equips SS with a remarkably effective foundation for solving a wide range of practical problems. The successes documented by Scatter Search come not so much from the adoption of adaptive memory in the range of ways proposed in Tabu Search (except where, as often happens, SS is advantageously coupled with TS), but from the use of strategic ideas initially proposed for exploiting adaptive memory, which blend harmoniously with the structure of Scatter Search. From a historical perspective, the dedicated use of heuristic strategies both to guide the process of combining solutions and to enhance the quality of offspring has been heralded as a key innovation in evolutionary methods, giving rise to what are sometimes called "hybrid" (or "memetic") evolutionary procedures. The underlying processes have been introduced into the mainstream of evolutionary methods (such as genetic algorithms, for example) by a series of gradual steps beginning in the late 1980s.

Geometric dynamics is a tool for developing a mathematical representation of real world phenomena, based on the notion of a field line described in two ways: -as the solution of any Cauchy problem associated to a first-order autonomous differential system; -as the solution of a certain Cauchy problem associated to a second-order conservative prolongation of the initial system. The basic novelty of our book is the discovery that a field line is a geodesic of a suitable geometrical structure on a given space (Lorentz-Udri~te world-force law). In other words, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds, ensuring that all trajectories of a given vector field are geodesics. This is our contribution to an old open problem studied by H. Poincare, S. Sasaki and others. From the kinematic viewpoint of corpuscular intuition, a field line shows the trajectory followed by a particle at a point of the definition domain of a vector field, if the particle is sensitive to the related type of field. Therefore, field lines appear in a natural way in problems of theoretical mechanics, fluid mechanics, physics, thermodynamics, biology, chemistry, etc.

Financial globalization has increased the significance of methods used in the evaluation of country risk, one of the major research topics in economics and finance. Written by experts in the fields of multicriteria methodology, credit risk assessment, operations research, and financial management, this book develops a comprehensive framework for evaluating models based on several classification techniques that emerge from different theoretical directions. This book compares different statistical and data mining techniques, noting the advantages of each method, and introduces new multicriteria methodologies that are important to country risk modeling.

Key topics include: (1) A review of country risk definitions and an overview of the most recent tools in country risk management, (2) In-depth analysis of statistical, econometric and non-parametric classification techniques, (3) Several real-world applications of the methodologies described throughout the text, (4) Future research directions for country risk assessment problems.

This work is a useful toolkit for economists, financial managers, bank managers, operations researchers, management scientists, and risk analysts. Moreover, the book can also be used as a supplementary text for graduate courses in finance and financial risk management.

In Linear Programming: A Modern Integrated Analysis, both boundary (simplex) and interior point methods are derived from the complementary slackness theorem and, unlike most books, the duality theorem is derived from Farkas's Lemma, which is proved as a convex separation theorem. The tedium of the simplex method is thus avoided. A new and inductive proof of Kantorovich's Theorem is offered, related to the convergence of Newton's method. Of the boundary methods, the book presents the (revised) primal and the dual simplex methods. An extensive discussion is given of the primal, dual and primal-dual affine scaling methods. In addition, the proof of the convergence under degeneracy, a bounded variable variant, and a super-linearly convergent variant of the primal affine scaling method are covered in one chapter. Polynomial barrier or path-following homotopy methods, and the projective transformation method are also covered in the interior point chapter. Besides the popular sparse Cholesky factorization and the conjugate gradient method, new methods are presented in a separate chapter on implementation. These methods use LQ factorization and iterative techniques.

"Mathematical Optimization and Economic Analysis" is a self-contained introduction to various optimization techniques used in economic modeling and analysis such as geometric, linear, and convex programming and data envelopment analysis. Through a systematic approach, this book demonstrates the usefulness of these mathematical tools in quantitative and qualitative economic analysis.

The book presents specific examples to demonstrate each technique's advantages and applicability as well as numerous applications of these techniques to industrial economics, regulatory economics, trade policy, economic sustainability, production planning, and environmental policy.

Key Features include:

- A detailed presentation of both single-objective and multiobjective optimization;

- An in-depth exposition of various applied optimization problems;

- Implementation of optimization tools to improve the accuracy of various economic models;

- Extensive resources suggested for further reading.

This book is intended for graduate and postgraduate students studying quantitative economics, as well as economics researchers and applied mathematicians. Requirements include a basic knowledge of calculus and linear algebra, and a familiarity with economic modeling.

Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have been described in many publications. In this monograph we try to present the methods in a consolidated way. We introduce several classes of operators, examine their properties, define iterative methods generated by operators from these classes and present general convergence theorems. On this basis we discuss the conditions under which particular methods converge. A large part of the results presented in this monograph can be found in various forms in the literature (although several results presented here are new). We have tried, however, to show that the convergence of a large class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems.

This book deals with a very important problem in power system planning for countries in which hydrogeneration accounts for the greatest part of the system power production. During the past thirty years many techniques have been developed to cope with the long-term operation of hydro reser voirs. These techniques have been discussed in a number of publications, but they have not until now been documented in book form. This book is intended as the foundation for a special graduate course dealing with aspects of electrical engineering, operational research, water resource research, and applied mathematics. It may also be used for self study by practicing personnel involved in the planning and operation of hydroelectric power systems for utilities, consulting groups, and government regulatory agencies. The book consists of eight chapters. Chapter 1 reviews the historical developments in the field, discusses briefly all techniques used to solve the problem, and summarizes the modeling of hydroplants for long-term operation studies. At the end of the chapter we present in detail an outline of the book."

Many kinds of practical problems such as engineering design, industrial m- agement and ?nancial investment have multiple objectives con?icting with eachother. Thoseproblemscanbeformulatedasmultiobjectiveoptimization. In multiobjective optimization, there does not necessarily a unique solution which minimizes (or maximizes) all objective functions. We usually face to the situation in which if we want to improve some of objectives, we have to give up other objectives. Finally, we pay much attention on how much to improve some of objectives and instead how much to give up others. This is called "trade-o?. " Note that making trade-o? is a problem of value ju- ment of decision makers. One of main themes of multiobjective optimization is how to incorporate value judgment of decision makers into decision s- port systems. There are two major issues in value judgment (1) multiplicity of value judgment and (2) dynamics of value judgment. The multiplicity of value judgment is treated as trade-o? analysis in multiobjective optimi- tion. On the other hand, dynamics of value judgment is di?cult to treat. However, it is natural that decision makers change their value judgment even in decision making process, because they obtain new information during the process. Therefore, decision support systems are to be robust against the change of value judgment of decision makers. To this aim, interactive p- grammingmethodswhichsearchasolutionwhileelicitingpartialinformation on value judgment of decision makers have been developed. Those methods are required to perform ?exibly for decision makers' attitude.

grams of which the objective is given by the ratio of a convex by a positive (over a convex domain) concave function. As observed by Sniedovich (Ref. [102, 103]) most of the properties of fractional pro grams could be found in other programs, given that the objective function could be written as a particular composition of functions. He called this new field C programming, standing for composite concave programming. In his seminal book on dynamic programming (Ref. [104]), Sniedovich shows how the study of such com positions can help tackling non-separable dynamic programs that otherwise would defeat solution. Barros and Frenk (Ref. [9]) developed a cutting plane algorithm capable of optimizing C-programs. More recently, this algorithm has been used by Carrizosa and Plastria to solve a global optimization problem in facility location (Ref. [16]). The distinction between global optimization problems (Ref. [54]) and generalized convex problems can sometimes be hard to establish. That is exactly the reason why so much effort has been placed into finding an exhaustive classification of the different weak forms of convexity, establishing a new definition just to satisfy some desirable property in the most general way possible. This book does not aim at all the subtleties of the different generalizations of convexity, but concentrates on the most general of them all, quasiconvex programming. Chapter 5 shows clearly where the real difficulties appear.

'Optimization Day' (OD) has been a series of annual mini-conferences in Aus tralia since 1994. The purpose of this series of events is to gather researchers in optimization and its related areas from Australia and their collaborators, in order to exchange new developments of optimization theories, methods and their applications. The first four OD mini-conferences were held in The Uni versity of Ballarat (1994), The University of New South Wales (1995), The University of Melbourne (1996) and Royal Melbourne Institute of Technology (1997), respectively. They were all on the eastern coast of Australia. The fifth mini-conference Optimization Days was held at the Centre for Ap plied Dynamics and Optimization (CADO), Department of Mathematics and Statistics, The University of Western Australia, Perth, from 29 to 30 June 1998. This is the first time the OD mini-conference has been held at the west ern coast of Australia. This fifth OD preceded the International Conference on Optimization: Techniques and Applications (ICOTA) held at Curtin Uni versity of Technology. Many participants attended both events. There were 28 participants in this year's mini-conference and 22 presentations in the mini conference. The presentations in this volume are refereed contributions based on papers presented at the fifth Optimization Days mini-conference. The volume is di vided into the following parts: Global Optimization, Nonsmooth Optimization, Optimization Methods and Applications."

This book reviews and discusses recent advances in the development of methods and algorithms for nonlinear optimization and its applications, focusing on the large-dimensional case, the current forefront of much research. Individual chapters, contributed by eminent authorities, provide an up-to-date overview of the field from different and complementary standpoints, including theoretical analysis, algorithmic development, implementation issues and applications.

This book contains a selection of refereed papers presented at the "International Conference on Operations Research (OR 2011)" which took place at the University of Zurich from August 30 to September 2, 2011. The conference was jointly organized by the German speaking OR societies from Austria (OGOR), Germany (GOR) and Switzerland (SVOR) under the patronage of SVOR. More than 840 scientists and students from over 50 countries attended OR 2011 and presented 620 papers in 16 parallel topical streams, as well as special award sessions. The conference was designed according to the understanding of Operations Research as an interdisciplinary science focusing on modeling complex socio-technical systems to gain insight into behavior under interventions by decision makers. Dealing with "organized complexity" lies in the core of OR and designing useful support systems to master the challenge of system management in complex environment is the ultimate goal of our professional societies. To this end, algorithmic techniques and system modeling are two fundamental competences which are also well-balanced in these proceedings."

After the IUTAM Symposium on Optimization in Structural Design held in Warsaw in 1973, it was clear to me that the time had come for organizing into a consistent body of thought the enormous quantity of results obtained in this domain, studied from so many different points of view, with so many different methods, and at so many levels of practical applicability. My colleague and friend Gianantonnio Sacchi from Milan and I met with Professor Prager in Savognin in July 1974, where I submitted to them my first ideas for a treatise on structural optimization: It should cover the whole domain from basic theory to practical applications, and deal with various materials, various types of structures, various functions required of the structures, and various types of cost . . Obviously, this was to be a team effort, to total three or four volumes, to be written in a balanced manner as textbooks and handbooks. Nothing similar existed at that time, and, indeed, nothing has been published to date. Professor Prager was immedi ately in favor of such a project. He agreed to write a first part on optimality criteria with me and to help me in the general organization of the series. Since Professor Sacchi was willing to write the text on variational methods, it remained to find authors for parts on the mathematical programming approach to structural optimization (and, more generally, on numerical methods) and on practical optimal design procedures in metal and concrete."

Engineering systems operate through actuators, most of which will exhibit phenomena such as saturation or zones of no operation, commonly known as dead zones. These are examples of piecewise-affine characteristics, and they can have a considerable impact on the stability and performance of engineering systems. This book targets controller design for piecewise affine systems, fulfilling both stability and performance requirements. The authors present a unified computational methodology for the analysis and synthesis of piecewise affine controllers, taking an approach that is capable of handling sliding modes, sampled-data, and networked systems. They introduce algorithms that will be applicable to nonlinear systems approximated by piecewise affine systems, and they feature several examples from areas such as switching electronic circuits, autonomous vehicles, neural networks, and aerospace applications. Piecewise Affine Control: Continuous-Time, Sampled-Data, and Networked Systems is intended for graduate students, advanced senior undergraduate students, and researchers in academia and industry. It is also appropriate for engineers working on applications where switched linear and affine models are important.

The point of departure in the present book is that the decision makers, involved in the evaluation of alternatives under conflicting criteria, express their preferential judgement by estimating ratios of subjective values or differences of the corresponding logarithms, the so-called grades. Three MCDA methods are studied in detail: the Simple Multi-Attribute Rating Technique SMART, as well as the Additive and the Multiplicative AHP, both pairwise-comparison methods which do not suffer from the well-known shortcomings of the original Analytic Hierarchy Process. Context-related preference modelling on the basis of psycho-physical research in visual perception and motor skills is extensively discussed in the introductory chapters. Thereafter many extensions of the ideas are presented via case studies in university administration, health care, environmental assessment, budget allocation, and energy planning at the national and the European level. The issues under consideration are: group decision making with inhomogeneous power distributions, the search for a compromise solution, resource allocation and fair distributions, scenario analysis in long-term planning, conflict analysis via the pairwise comparison of concessions, and multi-objective optimization. The final chapters are devoted to the fortunes of MCDA in the hands of its designers. The research started in the late seventies, when I got involved in three different problems: the nomination procedures in a university, the evaluation of alternative energy-research proposals, and the evaluation of non-linear programming software.

This book constitutes the refereed proceedings of the International Conference on Mathematical Modelling and Scientific Intelligence, ICMMSC 2012, Gandhigram, Tamil Nadu, India, in March 2012. The 62 revised full papers presented were carefully reviewed and selected from 332 submissions. The papers are organized in two topical sections on mathematical modelling and on scientific computation.

The Subject A little explanation is in order for our choice of the title Linear Opti- 1 mization (and corresponding terminology) for what has traditionally been called Linear Programming.Theword programming in this context can be confusing and/or misleading to students. Linear programming problems are referred to as optimization problems but the general term linear p- gramming remains. This can cause people unfamiliar with the subject to think that it is about programming in the sense of writing computer code. It isn't. This workbook is about the beautiful mathematics underlying the ideas of optimizing linear functions subject to linear constraints and the algorithms to solve such problems. In particular, much of what we d- cuss is the mathematics of Simplex Algorithm for solving such problems, developed by George Dantzig in the late 1940s. The word program in linear programming is a historical artifact. When Dantzig ?rstdevelopedthe Simplex Algorithm to solvewhat arenowcalled linear programming problems, his initial model was a class of resource - location problems to be solved for the U.S. Air Force. The decisions about theallocationswerecalled'Programs'bytheAirForce, andhencetheterm.

This monograph provides an introduction to the theory of topologies defined on the closed subsets of a metric space, and on the closed convex subsets of a normed linear space as well. A unifying theme is the relationship between topology and set convergence on the one hand, and set functionals on the other. The text includes for the first time anywhere an exposition of three topologies that over the past ten years have become fundamental tools in optimization, one-sided analysis, convex analysis, and the theory of multifunctions: the Wijsman topology, the Attouch--Wets topology, and the slice topology. Particular attention is given to topologies on lower semicontinuous functions, especially lower semicontinuous convex functions, as associated with their epigraphs. The interplay between convex duality and topology is carefully considered and a chapter on set-valued functions is included. The book contains over 350 exercises and is suitable as a graduate text. This book is of interest to those working in general topology, set-valued analysis, geometric functional analysis, optimization, convex analysis and mathematical economics.

From the reviews: "Do you know M.Padberg's Linear Optimization and Extensions? ...] Now here is the continuation of it, discussing the solutions of all its exercises and with detailed analysis of the applications mentioned. Tell your students about it. ...] For those who strive for good exercises and case studies for LP this is an excellent volume." Acta Scientiarum Mathematicarum

The complexity of issues requiring rational decision making grows and thus such decisions are becoming more and more difficult, despite advances in methodology and tools for decision support and in other areas of research. Globalization, interlinks between environmental, industrial, social and political issues, and rapid speed of change all contribute to the increase of this complexity. Specialized knowledge about decision-making processes and their support is increasing, but a large spectrum of approaches presented in the literature is typically illustrated only by simple examples. Moreover, the integration of model-based decision support methodologies and tools with specialized model-based knowledge developed for handling real problems in environmental, engineering, industrial, economical, social and political activities is often not satisfactory. Therefore, there is a need to present the state of art of methodology and tools for development of model-based decision support systems, and illustrate this state by applications to various complex real-world decision problems. The monograph reports many years of experience of many researchers, who have not only contributed to the developments in operations research but also succeeded to integrate knowledge and craft of various disciplines into several modern decision support systems which have been applied to actual complex decision-making processes in various fields of policy making. The experience presented in this book will be of value to researchers and practitioners in various fields. The issues discussed in this book gain in importance with the development of the new era of the information society, where information, knowledge, and ways of processing them become a decisive part of human activities. The examples presented in this book illustrate how how various methods and tools of model-based decision support can actually be used for helping modern decision makers that face complex problems. Overview of the contents: The first part of this three-part book presents the methodological background and characteristics of modern decision-making environment, and the value of model-based decision support thus addressing current challenges of decision support. It also provides the methodology of building and analyzing mathematical models that represent underlying physical and economic processes, and that are useful for modern decision makers at various stages of decision making. These methods support not only the analysis of Pareto-efficient solutions that correspond best to decision maker preferences but also allow the use of other modeling concepts like soft constraints, soft simulation, or inverse simulation. The second part describes various types of tools that are used for the development of decision support systems. These include tools for modeling, simulation, optimization, tools supporting choice and user interfaces. The described tools are both standard, commercially available, and nonstandard, public domain or shareware software, which are robust enough to be used also for complex applications. All four environmental applications (regional water quality management, land use planning, cost-effective policies aimed at improving the European air quality, energy planning with environmental implications) presented in the third part of the book rely on many years of cooperation between the authors of the book with several IIASA's projects, and with many researchers from the wide IIASA network of collaborating institutions. All these applications are characterized by an intensive use of model-based decision support. Finally, the appendix contains a short description of some of the tools described in the book that are available from IIASA, free of charge, for research and educational purposes. The experiences reported in this book indicate that the development of DSSs for strategic environmental decision making should be a joint effort involving experts in the subject area, modelers, and decision support experts. For the other experiences discussed in this book, the authors stress the importance of good data bases, and good libraries of tools. One of the most important requirements is a modular structure of a DSS that enhances the reusability of system modules. In such modular structures, user interfaces play an important role. The book shows how modern achievements in mathematical programming and computer sciences may be exploited for supporting decision making, especially about strategic environmental problems. It presents the methodological background of various methods for model-based decision support and reviews methods and tools for model development and analysis. The methods and tools are amply illustrated with extensive applications. Audience: This book will be of interest to researchers and practitioners in the fields of model development and analysis, model-based decision analysis and support, (particularly in the environment, economics, agriculture, engineering, and negotiations areas) and mathematical programming. For understanding of some parts of the text a background in mathematics and operational research is required but several chapters of the book will be of value also for readers without such a background. The monograph is also suitable for use as a text book for courses on advanced (Master and Ph.D.) levels for programs on Operations Research, decision analysis, decision support and various environmental studies (depending on the program different parts of the book may be emphasized).

This edited book is dedicated to Professor N. U. Ahmed, a leading scholar and a renowned researcher in optimal control and optimization on the occasion of his retirement from the Department of Electrical Engineering at University of Ottawa in 1999. The contributions of this volume are in the areas of optimal control, non linear optimization and optimization applications. They are mainly the im proved and expanded versions of the papers selected from those presented in two special sessions of two international conferences. The first special session is Optimization Methods, which was organized by K. L. Teo and X. Q. Yang for the International Conference on Optimization and Variational Inequality, the City University of Hong Kong, Hong Kong, 1998. The other one is Optimal Control, which was organized byK. Teo and L. Caccetta for the Dynamic Control Congress, Ottawa, 1999. This volume is divided into three parts: Optimal Control; Optimization Methods; and Applications. The Optimal Control part is concerned with com putational methods, modeling and nonlinear systems. Three computational methods for solving optimal control problems are presented: (i) a regularization method for computing ill-conditioned optimal control problems, (ii) penalty function methods that appropriately handle final state equality constraints, and (iii) a multilevel optimization approach for the numerical solution of opti mal control problems. In the fourth paper, the worst-case optimal regulation involving linear time varying systems is formulated as a minimax optimal con trol problem."

Mathematical Programming has been of significant interest and relevance in engineering, an area that is very rich in challenging optimization problems. In particular, many design and operational problems give rise to nonlinear and mixed-integer nonlinear optimization problems whose modeling and solu tion is often nontrivial. Furthermore, with the increased computational power and development of advanced analysis (e. g. , process simulators, finite element packages) and modeling systems (e. g. , GAMS, AMPL, SPEEDUP, ASCEND, gPROMS), the size and complexity of engineering optimization models is rapidly increasing. While the application of efficient local solvers (nonlinear program ming algorithms) has become widespread, a major limitation is that there is often no guarantee that the solutions that are generated correspond to global optima. In some cases finding a local solution might be adequate, but in others it might mean incurring a significant cost penalty, or even worse, getting an incorrect solution to a physical problem. Thus, the need for finding global optima in engineering is a very real one. It is the purpose of this monograph to present recent developments of tech niques and applications of deterministic approaches to global optimization in engineering. The present monograph is heavily represented by chemical engi neers; and to a large extent this is no accident. The reason is that mathematical programming is an active and vibrant area of research in chemical engineering. This trend has existed for about 15 years.