This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical models to various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well.

This proceedings volume, the fifth in a series from the Combinatorial and Additive Number Theory (CANT) conferences, is based on talks from the 19th annual workshop, held online due to the COVID-19 pandemic. Organized every year since 2003 by the New York Number Theory Seminar at the CUNY Graduate Center, the workshops survey state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. The CANT 2021 meeting featured over a hundred speakers from North and South America, Europe, Asia, Australia, and New Zealand, and was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain peer-reviewed and edited papers on current topics in number theory. Topics featured in this volume include sumsets, minimal bases, Sidon sets, analytic and prime number theory, combinatorial and discrete geometry, numerical semigroups, and a survey of expansion, divisibility, and parity. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Since the volume may be of interest to a broad variety of people, it is arranged in parts that require different levels of mathematical background. Part I is written in a simple form and can be assessed by any computer-literate person interested in the application of visualization methods in decision making. This part will be of interest to specialists and students in various fields related to decision making including environmental studies, management, business, engineering, etc. In Part II computational methods are introduced in a relatively simple form. This part will be of interest to specialists and students in the field of applied optimization, operations research and computer science. Part III is written for specialists and students in applied mathematics interested in the theoretical basis of modern optimization. Due to this structure, the parts can be read independently. For example, students interested in environmental applications could restrict themselves to Part I and the Epilogue. In contrast, those who are interested in computational methods can skip Part I and read Part II only. Finally, specialists, who are interested in the theory of approximation of multi-dimensional convex sets or in estimation of disturbances of polyhedral sets, can read the corresponding chapters of Part III.

This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight into Fuzzy Logic.

Fuzzy Sets have been introduced by Lotfi Zadeh in 1965 and since then, they have been used in many applications. As a consequence, there is a vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. The main purpose of the present book is to reduce this gap by providing a theoretical introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory. Well-known applications, as for example fuzzy control, are also discussed in this book and placed on new ground, a theoretical foundation. Moreover, a few advanced chapters and several new results are included. These comprise, among others, a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types, that investigates their approximation capability by providing new error estimates.

"

This edited volume is devoted to the now-ubiquitous use of computational models across most disciplines of engineering and science, led by a trio of world-renowned researchers in the field. Focused on recent advances of modeling and optimization techniques aimed at handling computationally-expensive engineering problems involving simulation models, this book will be an invaluable resource for specialists (engineers, researchers, graduate students) working in areas as diverse as electrical engineering, mechanical and structural engineering, civil engineering, industrial engineering, hydrodynamics, aerospace engineering, microwave and antenna engineering, ocean science and climate modeling, and the automotive industry, where design processes are heavily based on CPU-heavy computer simulations. Various techniques, such as knowledge-based optimization, adjoint sensitivity techniques, and fast replacement models (to name just a few) are explored in-depth along with an array of the latest techniques to optimize the efficiency of the simulation-driven design process. High-fidelity simulation models allow for accurate evaluations of the devices and systems, which is critical in the design process, especially to avoid costly prototyping stages. Despite this and other advantages, the use of simulation tools in the design process is quite challenging due to associated high computational cost. The steady increase of available computational resources does not always translate into the shortening of the design cycle because of the growing demand for higher accuracy and necessity to simulate larger and more complex systems. For this reason, automated simulation-driven design-while highly desirable-is difficult when using conventional numerical optimization routines which normally require a large number of system simulations, each one already expensive.

Game Theory has been an area of rapid growth and substantial interest in economics and it has impacted upon all areas within economics. This text covers the main theory and techniques and gives particular emphasis to aspects that have been neglected, including co-operative games, experiments, and empirical studies. It provides a comprehensive and up-to-date introduction to the use of game theory in economics.

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 includes a collection of research articles presented at the "6th International Workshop on Hydro Scheduling in Competitive Electricity Markets". The workshop was a unique and intimate forum for researchers and practitioners to present state-of-the-art research and development concerning novel methodological findings, best practices and real-life applications of hydro scheduling. It also provided a platform for discussing the developments that are taking place in the industry, sharing different experiences and discussing future trends related to this area. This proceedings book is a collection of the most relevant, high-quality articles from the workshop. Discussing the state-of-the-art in the field of hydro scheduling, it is a valuable resource for a wide audience of researchers and practitioners in the field now and in the interesting and challenging times ahead.

Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor. This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and Hamilton-Jacobi methods. The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering. Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the first-year or second-year graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday.Besides his startling 1960 result of the proof of the Poincare conjecture for all dimensionsgreater than or equal to five, Smale's ground breaking contributions invarious fields in Mathematics have marked the second part of the 20th century andbeyond. Stephen Smale has done pioneering work in differential topology, globalanalysis, dynamical systems, nonlinear functional analysis, numerical analysis, theoryof computation and machine learning as well as applications in the physical andbiological sciences and economics. In sum, Stephen Smale has manifestly brokenthe barriers among the different fields of mathematics and dispelled some remainingprejudices. He is indeed a universal mathematician. Smale has been honoredwith several prizes and honorary degrees including, among others, the Fields Medal(1966), The Veblen Prize (1966), the National Medal of Science (1996) and theWolfPrize (2006/2007)."

In 1995 the Handbook of Global Optimization (first volume), edited by R. Horst, and P.M. Pardalos, was published. This second volume of the Handbook of Global Optimization is comprised of chapters dealing with modern approaches to global optimization, including different types of heuristics.

Topics covered in the handbook include various metaheuristics, such as simulated annealing, genetic algorithms, neural networks, taboo search, shake-and-bake methods, and deformation methods. In addition, the book contains chapters on new exact stochastic and deterministic approaches to continuous and mixed-integer global optimization, such as stochastic adaptive search, two-phase methods, branch-and-bound methods with new relaxation and branching strategies, algorithms based on local optimization, and dynamical search. Finally, the book contains chapters on experimental analysis of algorithms and software, test problems, and applications.

A state-of-the-art research monograph providing consistent treatment of supervisory control, by one of the world 's leading groups in the area of Bayesian identification, control, and decision making. An accompanying CD illustrates the book 's underlying theory.

This text provides the undergraduate chemical engineering student with the necessary tools for problem solving in chemical or bio-engineering processes. In a friendly, simple, and unified framework, the exposition aptly balances theory and practice. It uses minimal mathematical concepts, terms, algorithms, and describes the main aspects of chemical process optimization using MATLAB and GAMS. Numerous examples and case studies are designed for students to understand basic principles of each optimization method and elicit the immediate discovery of practical applications. Problem sets are directly tied to real-world situations most commonly encountered in chemical engineering applications. Chapters are structured with handy learning summaries, terms and concepts, and problem sets, and individually reinforce the basics of particular optimization methods. Additionally, the wide breadth of topics that may be encountered in courses such as Chemical Process Optimization, Chemical Process Engineering, Optimization of Chemical Processes, are covered in this accessible text. The book provides formal introductions to MATLAB, GAMS, and a revisit to pertinent aspects of undergraduate calculus. While created for coursework, this text is also suitable for independent study. A full solutions manual is available to instructors who adopt the text for their course.

Optimization problems abound in most fields of science, engineering, and tech nology. 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, many times satisfy certain constraints. Global optimization problems belong to the complexity class of NP-hard prob lems. Such problems are very difficult to solve. Traditional descent optimization algorithms based on local information are not adequate for solving these problems. In most cases of practical interest the number of local optima increases, on the aver age, 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 the past ten years, due to the success of new algorithms for solving large classes of problems from diverse areas such as engineering design and control, computational chemistry and biology, structural optimization, computer science, operations research, and economics. This book contains refereed invited 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 re search on global optimization and related applications in science and engineering. The papers included in this book cover a wide spectrum of approaches for solving global optimization problems and applications."

This book is an up-to-date documentation of the state of the art in combinatorial optimization, presenting approximate solutions of virtually all relevant classes of NP-hard optimization problems. The well-structured wealth of problems, algorithms, results, and techniques introduced systematically will make the book an indispensible source of reference for professionals. The smooth integration of numerous illustrations, examples, and exercises make this monograph an ideal textbook.

This book is a rigorous but practical presentation of the techniques of uncertainty quantification, with applications in R and Python. This volume includes mathematical arguments at the level necessary to make the presentation rigorous and the assumptions clearly established, while maintaining a focus on practical applications of uncertainty quantification methods. Practical aspects of applied probability are also discussed, making the content accessible to students. The introduction of R and Python allows the reader to solve more complex problems involving a more significant number of variables. Users will be able to use examples laid out in the text to solve medium-sized problems. The list of topics covered in this volume includes linear and nonlinear programming, Lagrange multipliers (for sensitivity), multi-objective optimization, game theory, as well as linear algebraic equations, and probability and statistics. Blending theoretical rigor and practical applications, this volume will be of interest to professionals, researchers, graduate and undergraduate students interested in the use of uncertainty quantification techniques within the framework of operations research and mathematical programming, for applications in management and planning.

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

Game theory now provides the theoretical underpinning for a multitude of courses in economics worldwide. The speed of these developments has been remarkable and they have constituted something of a revolution. Indeed, the basic tenets of game theory have now begun to colonize other social sciences and its proponents have been unified in claiming its natural basis as a rational theory of society.
But the technical demands of the subject and the lack of clarity of many commentators' explanations have been a problem for many. This second edition of Shaun Hargreaves-Heap and Yanis Varoufakis's classic text strips away the mystique and lets the student make his or her own mind up. This new edition:
* lays bear the philosophy behind game theory and explains all its major concepts
* conducts a clear explanation and careful analysis of all the main 'games', such as repeated games and the prisoner's dilemma, as well as less well-trodden topics such as experimental games and issues of rationality
* introduces and analyses new areas of game theory such as "evolutionary games" and "Psychological games"
Retaining the most popular features from the original edition, this latest version offers improved pedagogy, new questions for discussion, new chapters on the latest developments as well as a host of new examples. A new generation of students of game theory will grow to appreciate this superb text whilst fans of the first edition will eagerly devour this long-awaited update.

Bilevel programming problems are hierarchical optimization problems where the constraints of one problem (the so-called upper level problem) are defined in part by a second parametric optimization problem (the lower level problem). If the lower level problem has a unique optimal solution for all parameter values, this problem is equivalent to a one-level optimization problem having an implicitly defined objective function. Special emphasize in the book is on problems having non-unique lower level optimal solutions, the optimistic (or weak) and the pessimistic (or strong) approaches are discussed. The book starts with the required results in parametric nonlinear optimization. This is followed by the main theoretical results including necessary and sufficient optimality conditions and solution algorithms for bilevel problems. Stationarity conditions can be applied to the lower level problem to transform the optimistic bilevel programming problem into a one-level problem. Properties of the resulting problem are highlighted and its relation to the bilevel problem is investigated. Stability properties, numerical complexity, and problems having additional integrality conditions on the variables are also discussed. Audience: Applied mathematicians and economists working in optimization, operations research, and economic modelling. Students interested in optimization will also find this book useful.

This book introduces a holistic approach to ship design and its optimisation for life-cycle operation. It deals with the scientific background of the adopted approach and the associated synthesis model, which follows modern computer aided engineering (CAE) procedures. It integrates techno-economic databases, calculation and multi-objective optimisation modules and s/w tools with a well-established Computer-Aided Design (CAD) platform, along with a Virtual Vessel Framework (VVF), which will allow virtual testing before the building phase of a new vessel. The resulting graphic user interface (GUI) and information exchange systems enable the exploration of the huge design space to a much larger extent and in less time than is currently possible, thus leading to new insights and promising new design alternatives. The book not only covers the various stages of the design of the main ship system, but also addresses relevant major onboard systems/components in terms of life-cycle performance to offer readers a better understanding of suitable outfitting details, which is a key aspect when it comes the outfitting-intensive products of international shipyards. The book disseminates results of the EU funded Horizon 2020 project HOLISHIP.

Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aim to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: - On the complexity of combinatorial optimization problems, presenting basics about worst-case and randomized complexity; - Classical solution methods, presenting the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; - Elements from mathematical programming, presenting fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.

This book presents the best papers from the 3rd International Conference on Mathematical Research for Blockchain Economy (MARBLE) 2022, held in Vilamoura, Portugal. While most blockchain conferences and forums are dedicated to business applications, product development or Initial Coin Offering (ICO) launches, this conference focuses on the mathematics behind blockchain to bridge the gap between practice and theory. Blockchain Technology has been considered as the most fundamental and revolutionising invention since the Internet. Every year, thousands of blockchain projects are launched and circulated in the market, and there is a tremendous wealth of blockchain applications, from finance to healthcare, education, media, logistics and more. However, due to theoretical and technical barriers, most of these applications are impractical for use in a real-world business context. The papers in this book reveal the challenges and limitations, such as scalability, latency, privacy and security, and showcase solutions and developments to overcome them.

This volume contains, in part, a selection of papers presented at the sixth Australian Optimization Day Miniconference (Ballarat, 16 July 1999), and the Special Sessions on Nonlinear Dynamics and Optimization and Operations Re search - Methods and Applications, which were held in Melbourne, July 11-15 1999 as a part of the Joint Meeting of the American Mathematical Society and Australian Mathematical Society. The editors have strived to present both con tributed papers and survey style papers as a more interesting mix for readers. Some participants from the meetings mentioned above have responded to this approach by preparing survey and 'semi-survey' papers, based on presented lectures. Contributed paper, which contain new and interesting results, are also included. The fields of the presented papers are very large as demonstrated by the following selection of key words from selected papers in this volume: * optimal control, stochastic optimal control, MATLAB, economic models, implicit constraints, Bellman principle, Markov process, decision-making under uncertainty, risk aversion, dynamic programming, optimal value function. * emergent computation, complexity, traveling salesman problem, signal estimation, neural networks, time congestion, teletraffic. * gap functions, nonsmooth variational inequalities, derivative-free algo rithm, Newton's method. * auxiliary function, generalized penalty function, modified Lagrange func tion. * convexity, quasiconvexity, abstract convexity.

This book provides an introduction to the applications of game theory to a series of questions that are fundamental in political economy. These questions include: Why do we need states? What might happen without protection for life and property? How might tribes or criminal gangs behave in struggles over material possessions? Would people tell the truth if asked what they wanted?

Although the monograph Progress in Optimization I: Contributions from Aus tralasia grew from the idea of publishing a proceedings of the Fourth Optimiza tion Day, held in July 1997 at the Royal Melbourne Institute of Technology, the focus soon changed to a refereed volume in optimization. The intention is to publish a similar book annually, following each Optimization Day. The idea of having an annual Optimization Day was conceived by Barney Glover; the first of these Optimization Days was held in 1994 at the University of Ballarat. Barney hoped that such a yearly event would bring together the many, but widely dispersed, researchers in Australia who were publishing in optimization and related areas such as control. The first Optimization Day event was followed by similar conferences at The University of New South Wales (1995), The University of Melbourne (1996), the Royal Melbourne Institute of Technology (1997), and The University of Western Australia (1998). The 1999 conference will return to Ballarat University, being organized by Barney's long-time collaborator Alex Rubinov. In recent years the Optimization Day has been held in conjunction with other locally-held national or international conferences. This has widened the scope of the monograph with contributions not only coming from researchers in Australia and neighboring regions but also from their collaborators in Europe and North America."