Nonconvex Optimization and Its Applications – serie

Studies in generalized convexity and generalized monotonicity have significantly increased during the last two decades. Researchers with very diverse backgrounds such as mathematical programming, optimization theory, convex analysis, nonlinear analysis, nonsmooth analysis, linear algebra, probability theory, variational inequalities, game theory, economic theory, engineering, management science, equilibrium analysis, for example are attracted to this fast growing field of study. Such enormous research activity is partially due to the discovery of a rich, elegant and deep theory which provides a basis for interesting existing and potential applications in different disciplines. The handbook offers an advanced and broad overview of the current state of the field. It contains fourteen chapters written by the leading experts on the respective subject; eight on generalized convexity and the remaining six on generalized monotonicity.

In recent years there is a growing interest in generalized convex fu- tions and generalized monotone mappings among the researchers of - plied mathematics and other sciences. This is due to the fact that mathematical models with these functions are more suitable to describe problems of the real world than models using conventional convex and monotone functions. Generalized convexity and monotonicity are now considered as an independent branch of applied mathematics with a wide range of applications in mechanics, economics, engineering, finance and many others. The present volume contains 20 full length papers which reflect c- rent theoretical studies of generalized convexity and monotonicity, and numerous applications in optimization, variational inequalities, equil- rium problems etc. All these papers were refereed and carefully selected from invited talks and contributed talks that were presented at the 7th International Symposium on Generalized Convexity/Monotonicity held in Hanoi, Vietnam, August 27-31, 2002. This series of Symposia is or- nized by the Working Group on Generalized Convexity (WGGC) every 3 years and aims to promote and disseminate research on the field. The WGGC (http://www.genconv.org) consists of more than 300 researchers coming from 36 countries.

This Volume contains the (refereed) papers presented at the 38th Conference of the School of Mathematics "G.Stampacchia" of the "E.Majorana" Centre for Scientific Culture of Erice (Sicily), held in Memory ofG. Stampacchia and J.-L. Lions in the period June 20 - July 2003. The presence of participants from Countries has greatly contributed to the success of the meeting. The School of Mathematics was dedicated to Stampacchia, not only for his great mathematical achievements, but also because He founded it. The core of the Conference has been the various features of the Variational Analysis and their motivations and applications to concrete problems. Variational Analysis encompasses a large area of modem Mathematics, such as the classical Calculus of Variations, the theories of perturbation, approximation, subgradient, subderivates, set convergence and Variational Inequalities, and all these topics have been deeply and intensely dealt during the Conference. In particular, Variational Inequalities, which have been initiated by Stampacchia, inspired by Signorini Problem and the related work of G. Fichera, have offered a very great possibility of applications to several fundamental problems of Mathematical Physics, Engineering, Statistics and Economics. The pioneer work of Stampacchia and Lions can be considered as the basic kernel around which Variational Analysis is going to be outlined and constructed. The Conference has dealt with both finite and infinite dimensional analysis, showing that to carry on these two aspects disjointly is unsuitable for both.

Quadratic programs and affine variational inequalities represent two fundamental, closely-related classes of problems in the t,heories of mathematical programming and variational inequalities, resp- tively. This book develops a unified theory on qualitative aspects of nonconvex quadratic programming and affine variational inequ- ities. The first seven chapters introduce the reader step-by-step to the central issues concerning a quadratic program or an affine variational inequality, such as the solution existence, necessary and sufficient conditions for a point to belong to the solution set, and properties of the solution set. The subsequent two chapters discuss briefly two concrete nlodels (linear fractional vector optimization and the traffic equilibrium problem) whose analysis can benefit a lot from using the results on quadratic programs and affine variational inequalities. There are six chapters devoted to the study of conti- ity and/or differentiability properties of the characteristic maps and functions in quadratic programs and in affine variational inequa- ties where all the components of the problem data are subject to perturbation. Quadratic programs and affine variational inequa- ties under linear perturbations are studied in three other chapters. One special feature of the presentation is that when a certain pr- erty of a characteristic map or function is investigated, we always try first to establish necessary conditions for it to hold, then we go on to study whether the obtained necessary conditions are suf- cient ones. This helps to clarify the structures of the two classes of problems under consideration.

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.

Most books about global optimization describe the theory of the algorithms, whereas a given implementation’s quality never depends exclusively on the theoretical soundness of the algorithms that are implemented. The literature rarely discusses the tuning of algorithmic parameters, implementation tricks, software architectures, and the embedding of local solvers within global solvers. And yet, there are many good software implementations "out there” from which the entire community could learn something.The scope of this book is moving a few steps toward the systematization of the path that goes from the invention to the implementation and testing of a global optimization algorithm. Some of the contributors to the book are famous and some are less well-known, but all are experts in the discipline of actually getting global optimization to work. Thus, the papers in this book address the following topics:- Descriptions of new implementations of general-purpose or problem-specific global optimization algorithms- New algorithms in global optimization (some with numerical results and a discussion of the implementation)- Surveys discussing existing software packages.

Robust design—that is, managing design uncertainties such as model uncertainty or parametric uncertainty—is the often unpleasant issue crucial in much multidisciplinary optimal design work. Recently, there has been enormous practical interest in strategies for applying optimization tools to the development of robust solutions and designs in several areas, including aerodynamics, the integration of sensing (e.g., laser radars, vision-based systems, and millimeter-wave radars) and control, cooperative control with poorly modeled uncertainty, cascading failures in military and civilian applications, multi-mode seekers/sensor fusion, and data association problems and tracking systems. The contributions to this book explore these different strategies. The expression "optimization-directed” in this book’s title is meant to suggest that the focus is not agonizing over whether optimization strategies identify a true global optimum, but rather whether these strategies make significant design improvements.

As optimization researchers tackle larger and larger problems, scale interactions play an increasingly important role. One general strategy for dealing with a large or difficult problem is to partition it into smaller ones, which are hopefully much easier to solve, and then work backwards towards the solution of original problem, using a solution from a previous level as a starting guess at the next level. This volume contains 22 chapters highlighting some recent research. The topics of the chapters selected for this volume are focused on the development of new solution methodologies, including general multilevel solution techniques, for tackling difficult, large-scale optimization problems that arise in science and industry. Applications presented in the book include but are not limited to the circuit placement problem in VLSI design, a wireless sensor location problem, optimal dosages in the treatment of cancer by radiation therapy, and facility location.

Large-Scale Nonlinear Optimization 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.The chapters of the book, authored by some of the most active and well-known researchers in nonlinear optimization, give an updated overview of the field from different and complementary standpoints, including theoretical analysis, algorithmic development, implementation issues and applications.

Optimization models based on a nonlinear systems description often possess multiple local optima. The objective of global optimization (GO) is to find the best possible solution of multiextremal problems. This volume illustrates the applicability of GO modeling techniques and solution strategies to real-world problems. The contributed chapters cover a broad range of applications from agroecosystem management, assembly line design, bioinformatics, biophysics, black box systems optimization, cellular mobile network design, chemical process optimization, chemical product design, composite structure design, computational modeling of atomic and molecular structures, controller design for induction motors, electrical engineering design, feeding strategies in animal husbandry, the inverse position problem in kinematics, laser design, learning in neural nets, mechanical engineering design, numerical solution of equations, radiotherapy planning, robot design, and satellite data analysis. The solution strategies discussed encompass a range of practically viable methods, including both theoretically rigorous and heuristic approaches.

This book is devoted to the study and optimization of spatiotem- ral stochastic processes. These are processes that simultaneously - velop in space and time under random influences. Such processes - cur almost everywhere when the global behavior of complex systems is studied, e.g., in physical and technical systems, population dynamics, neural networks, computer and telecommunication networks, complex production networks and flexible manufacturing systems, logistic n- works and transportation systems, environmental engineering, climate modeling and prediction, earth surface models, and so on. In the study of spatiotemporal stochastic processes the classical c- cepts of random fields (which are models for spatially distributed r- dom phenomena) and of stochastic processes (hich are usually thought to describe the evolution over time of systems under random influences) converge. Over the last twenty years,many research monographs were written with emphasis on this unifying point of view, as were a huge number of articles and papers on this subject.

Complementarity theory, a relatively new domain in applied mathematics, has deep connections with several aspects of fundamental mathematics and also has many applications in optimization, economics and engineering. The study of variational inequalities is another domain of applied mathematics with many applications to the study of certain problems with unilateral conditions. This book is the first to discuss complementarity theory and variational inequalities using Leray–Schauder type alternatives. The ideas and method presented in this book may be considered as a starting point for new developments.

2. The Algorithm ...59 3. Convergence Analysis ..., ...60 4. Complexity Analysis ...63 5. Conclusions ...67 References ...67 A Simple Proof for a Result of Ollerenshaw on Steiner Trees ...68 Xiufeng Du, Ding-Zhu Du, Biao Gao, and Lixue Qii 1. Introduction ...68 2. In the Euclidean Plane ...69 3. In the Rectilinear Plane ...70 4. Discussion ...-...71 References ...71 Optimization Algorithms for the Satisfiability (SAT) Problem ...72 Jun Gu 1. Introduction ...72 2. A Classification of SAT Algorithms ...7:3 3. Preliminaries ...IV 4. Complete Algorithms and Incomplete Algorithms ...81 5. Optimization: An Iterative Refinement Process ...86 6. Local Search Algorithms for SAT ...89 7. Global Optimization Algorithms for SAT Problem ...106 8. Applications ...137 9. Future Work ...140 10. Conclusions ...141 References ...143 Ergodic Convergence in Proximal Point Algorithms with Bregman Functions ...155 Osman Guier 1. Introduction ...: ...155 2. Convergence for Function Minimization ...158 3. Convergence for Arbitrary Maximal Monotone Operators ...161 References ...163 Adding and Deleting Constraints in the Logarithmic Barrier Method for LP ...166 D. den Hertog, C. Roos, and T. Terlaky 1.Introduction ...16(5 2. The Logarithmic Darrier Method ...lG8 CONTENTS IX 3. The Effects of Shifting, Adding and Deleting Constraints ...171 4. The Build-Up and Down Algorithm ...177 ...5. Complexity Analysis ...180 References ...184 A Projection Method for Solving Infinite Systems of Linear Inequalities ...186 Hui Hu 1. Introduction ...186 2. The Projection Method ...186 3. Convergence Rate ...189 4. Infinite Systems of Convex Inequalities ...191 5. Application ...193 References ...

Global optimization is concerned with the computation and characterization of global optima of nonlinear functions. During the past three decades the field of global optimization has been growing at a rapid pace, and the number of publications on all aspects of global optimization has been increasing steadily. Many applications, as well as new theoretical, algorithmic, and computational contributions have resulted. The Handbook of Global Optimization is the first comprehensive book to cover recent developments in global optimization. Each contribution in the Handbook is essentially expository in nature, but scholarly in its treatment. The chapters cover optimality conditions, complexity results, concave minimization, DC programming, general quadratic programming, nonlinear complementarity, minimax problems, multiplicative programming, Lipschitz optimization, fractional programming, network problems, trajectory methods, homotopy methods, interval methods, and stochastic approaches. The Handbook of Global Optimization is addressed to researchers in mathematical programming, as well as all scientists who use optimization methods to model and solve problems.

Global optimization concerns the computation and characterization of global optima of nonlinear functions. Such problems are widespread in the mathematical modelling of real systems in a very wide range of applications and the last 30 years have seen the development of many new theoretical, algorithmic and computational contributions which have helped to solve globally multiextreme problems in important practical applications. Most of the existing books on optimization focus on the problem of computing locally optimal solutions. Introduction to Global Optimization, however, is a comprehensive textbook on constrained global optimization that covers the fundamentals of the subject, presenting much new material, including algorithms, applications and complexity results for quadratic programming, concave minimization, DC and Lipschitz problems, and nonlinear network flow. Each chapter contains illustrative examples and ends with carefully selected exercises, designed to help students grasp the material and enhance their knowledge of the methods involved. Audience: Students of mathematical programming, and all scientists, from whatever discipline, who need global optimization methods in such diverse areas as economic modelling, fixed charges, finance, networks and transportation, databases, chip design, image processing, nuclear and mechanical design, chemical engineering design and control, molecular biology, and environmental engineering.

Global optimization concerns the computation and characterization of global optima of nonlinear functions. Such problems are widespread in the mathematical modelling of real systems in a very wide range of applications and the last 30 years have seen the development of many new theoretical, algorithmic and computational contributions which have helped to solve globally multiextreme problems in important practical applications. Most of the existing books on optimization focus on the problem of computing locally optimal solutions. Introduction to Global Optimization, however, is a comprehensive textbook on constrained global optimization that covers the fundamentals of the subject, presenting much new material, including algorithms, applications and complexity results for quadratic programming, concave minimization, DC and Lipschitz problems, and nonlinear network flow. Each chapter contains illustrative examples and ends with carefully selected exercises, designed to help students grasp the material and enhance their knowledge of the methods involved. Audience: Students of mathematical programming, and all scientists, from whatever discipline, who need global optimization methods in such diverse areas as economic modelling, fixed charges, finance, networks and transportation, databases, chip design, image processing, nuclear and mechanical design, chemical engineering design and control, molecular biology, and environmental engineering.

Techniques and principles of minimax theory play a key role in many areas of research, including game theory, optimization, and computational complexity. In general, a minimax problem can be formulated as min max f(x, y) (1) ",EX !lEY where f(x, y) is a function defined on the product of X and Y spaces. There are two basic issues regarding minimax problems: The first issue concerns the establishment of sufficient and necessary conditions for equality minmaxf(x,y) = maxminf(x,y). (2) "'EX !lEY !lEY "'EX The classical minimax theorem of von Neumann is a result of this type. Duality theory in linear and convex quadratic programming interprets minimax theory in a different way. The second issue concerns the establishment of sufficient and necessary conditions for values of the variables x and y that achieve the global minimax function value f(x*, y*) = minmaxf(x, y). (3) "'EX !lEY There are two developments in minimax theory that we would like to mention.

As its title implies, Advances in Multicriteria Analysis presents the most recent developments in multicriteria analysis and in some of its principal areas of application, including marketing, research and development evaluation, financial planning, and medicine. Special attention is paid to the interaction between multicriteria analysis, decision support systems and preference modeling. The five sections of the book cover: methodology; problem structuring; utility assessment; multi-objective optimisation; real world applications. Audience: Researchers and professionals who are operations researchers, management scientists, computer scientists, statisticians, decision analysts, marketing managers and financial analysts.

In science, engineering and economics, decision problems are frequently modelled by optimizing the value of a (primary) objective function under stated feasibility constraints. In many cases of practical relevance, the optimization problem structure does not warrant the global optimality of local solutions; hence, it is natural to search for the globally best solution(s). This study provides a discussion of adaptive partition strategies to solve global optimization problems under very general structural requirements. A unified approach to numerous known algorithms makes possible straightforward generalizations and extensions, leading to efficient computer-based implementations. A considerable part of the book is devoted to applications, including some generic problems from numerical analysis, and several case studies in environmental systems analysis and management.

Optimization problems abound in most fields of science, engineering and technology. In many of these problems it is necessary to compute the global optimum (or a good approximation) of a multivariable function. The variables that define the function to be optimized can be continuous and/or discrete and, in addition, they often have to satisfy certain constraints. Global optimization problems belong to the complexity class of NP-hard problems. Such problems are difficult to solve; traditional descent optimization algorithms based on local information are inadequate for solving them. In most cases of practical interest the number of local optima increases, on the average, exponentially with the size of the problem (number of variables). Furthermore, most of the traditional approaches fail to escape from a local optimum in order to continue the search for the global solution.Global optimization has received a lot of attention in recent years, due to the success of new algorithms for solving large classes of problems from diverse areas such as computational chemistry and biology, structural optimization, computer sciences, operations research, economics, and engineering design and control. This work contains papers presented at the conference on "State of the Art in Global Optimization: Computational Methods and Applications" held at Princeton University, April 28-30, 1995. The conference presented current research on global optimization and related applications in science and engineering. The papers included cover a wide spectrum of approaches for solving global optimization problems and applications.

Motivation Stochastic Linear Programming with recourse represents one of the more widely applicable models for incorporating uncertainty within in which the SLP optimization models. There are several arenas model is appropriate, and such models have found applications in air­ line yield management, capacity planning, electric power generation planning, financial planning, logistics, telecommunications network planning, and many more. In some of these applications, modelers represent uncertainty in terms of only a few seenarios and formulate a large scale linear program which is then solved using LP software. However, there are many applications, such as the telecommunications planning problem discussed in this book, where a handful of seenarios do not capture variability well enough to provide a reasonable model of the actual decision-making problem. Problems of this type easily exceed the capabilities of LP software by several orders of magnitude. Their solution requires the use of algorithmic methods that exploit the structure of the SLP model in a manner that will accommodate large scale applications.

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.

This text provides a rigorous presentation of the quasidifferentiability approach to nonconvex, possibly nonsmooth, energy functions of the derivation and study of the corresponding variational expressions in mechanics, engineering and economics, and of their numerical treatment. The new variational formulations derived are illustrated by many interesting numerical problems. The techniques presented should permit the reader to check any solution obtained by other heuristic techniques for nonconvex, nonsmooth energy problems. Civil, mechanical or aeronautical engineers should find in the book a mathematically sound technique for the formulation and study of nonconvex, nonsmooth energy problems.

axiomatic results should be at the heart of such a science. Through them, we should be able to enlighten and scientifically assist decision-making processes especially by: - making that wh ich is objective stand out more c1early from that which is less objective; - separating robust from fragile conc1usions; - dissipating certain forms of misunderstanding in communication; - avoiding the pitfall of illusory reasoning; - emphasizing, once they are understood, incontrovertible results. The difficulties I encountered at the begining of my career as an operations researcher, and later as a consultant, made me realize that there were some limitations on objectivity in decision-aiding. In my opinion, five major aspects must be taken into consideration: 1) The borderline (or frontier) between what is and what is not feasible is often fuzzy. Moreover, this borderline is frequently modified in light of what is found from the study itself. 2) In many real-world problems, the "decision maker D" does not really exist as a person truly able to make adecision. Usually, several people (actors or stakeholders) take part in the decision process, and it is important not to confuse the one who ratifies adecision with the so-called decision maker in the decision ai ding process. This decision maker is in fact the person or the set of persons for whom or in the name of whom decision aiding effort is provided.

This text provides a self-contained introduction to underlying techniques, as well as a compendium of theory and a guide to the author's Fortran 90 software for nonlinear algebraic systems and global, constrained optimization with automatic result verification. Besides introductory and survey material, the book contains research results. It also contains non-traditional ideas concerning non-smooth optimization. Thus, the book should be a valuable reference to applied mathematicians and computational scientists and engineers. With numerous examples and exercises, as well as leads for future research, the book can be used as a graduate text or reference on interval arithmetic, automatic differentiation and interval fixed point theory. The book can also serve as a user's guide for the nonlinear equations and optimization software, available free of charge from the author, for the author's general Fortran 90 interval arithmetic package, or for the associated automatic differentiation package.

This book deals with decision making in environments of significant data un­ certainty, with particular emphasis on operations and production management applications. For such environments, we suggest the use of the robustness ap­ proach to decision making, which assumes inadequate knowledge of the decision maker about the random state of nature and develops a decision that hedges against the worst contingency that may arise. The main motivating factors for a decision maker to use the robustness approach are: • It does not ignore uncertainty and takes a proactive step in response to the fact that forecasted values of uncertain parameters will not occur in most environments; • It applies to decisions of unique, non-repetitive nature, which are common in many fast and dynamically changing environments; • It accounts for the risk averse nature of decision makers; and • It recognizes that even though decision environments are fraught with data uncertainties, decisions are evaluated ex post with the realized data. For all of the above reasons, robust decisions are dear to the heart of opera­ tional decision makers. This book takes a giant first step in presenting decision support tools and solution methods for generating robust decisions in a variety of interesting application environments. Robust Discrete Optimization is a comprehensive mathematical programming framework for robust decision making.

Global optimization is one of the fastest developing fields in mathematical optimization. In fact, an increasing number of remarkably efficient deterministic algorithms have been proposed in the last ten years for solving several classes of large scale specially structured problems encountered in such areas as chemical engineering, financial engineering, location and network optimization, production and inventory control, engineering design, computational geometry, and multi-objective and multi-level optimization. These new developments motivated the authors to write a new book devoted to global optimization problems with special structures. Most of these problems, though highly nonconvex, can be characterized by the property that they reduce to convex minimization problems when some of the variables are fixed. A number of recently developed algorithms have been proved surprisingly efficient for handling typical classes of problems exhibiting such structures, namely low rank nonconvex structures. Audience: The book will serve as a fundamental reference book for all those who are interested in mathematical optimization.

Industrial, financial, commercial or any kinds of project have at least one common feature: the better organized they are, the higher the profit or the lower the cost. Project management is the principle of planning different projects and keeping them on track within time, cost and resource constraints. The need for effective project management is ever-increasing. The complexity of the environment we live in requires more sophisticated methods than it did just a couple of decades ago. Project managers might face insurmountable obstacles in their work if they do not adapt themselves to the changing circumstances. On the other hand, better knowledge of project management can result in better plans, schedules and, last but not least, more contracts and more profit. This knowledge can help individuals and firms to stay alive in this competitive market and, in the global sense, utilize the finite resources of our planet in a more efficient way.

Bayesian decision theory is known to provide an effective framework for the practical solution of discrete and nonconvex optimization problems. This book is the first to demonstrate that this framework is also well suited for the exploitation of heuristic methods in the solution of such problems, especially those of large scale for which exact optimization approaches can be prohibitively costly. The book covers all aspects ranging from the formal presentation of the Bayesian Approach, to its extension to the Bayesian Heuristic Strategy, and its utilization within the informal, interactive Dynamic Visualization strategy. The developed framework is applied in forecasting, in neural network optimization, and in a large number of discrete and continuous optimization problems. Specific application areas which are discussed include scheduling and visualization problems in chemical engineering, manufacturing process control, and epidemiology. Computational results and comparisons with a broad range of test examples are presented. The software required for implementation of the Bayesian Heuristic Approach is included. Although some knowledge of mathematical statistics is necessary in order to fathom the theoretical aspects of the development, no specialized mathematical knowledge is required to understand the application of the approach or to utilize the software which is provided. Audience: The book is of interest to both researchers in operations research, systems engineering, and optimization methods, as well as applications specialists concerned with the solution of large scale discrete and/or nonconvex optimization problems in a broad range of engineering and technological fields. It may be used as supplementary material for graduate level courses.