Richard Bellmann developed a theory of dynamic programming which is for many reasons still in the center of great interest. The authors present a new approach in the ?eld of the optimization and multi-objective control of time-discrete systems which is closely related to the work of Richard Bellmann. They develop their own concept and their extension to the optimization and multi-objective control of time-discrete systems as well as to dynamic networks and multilayered structures are very stimulating for further research. Di?erent perspectives of discrete control and optimal dynamic ?ow problems on networks are treated and characterized. Together with the algorithmic solutions a framework of multi-objective control problems is - rived. The conclusion with a real world example underlines the necessity and - portance of their theoretic framework. As they come back to the classical Bellmann concept of dynamic programming they stress and honor his basic concept without debase their own work. Multilayereddecisionprocessesaspartofthedesignandanalysisofcomplexsystems and networks will be essential in many ways and ?elds in the future.

Interest in regularization methods for ill-posed nonlinear operator equations and variational inequalities of monotone type in Hilbert and Banach spaces has grown rapidly over recent years. Results in the field over the last three decades, previously only available in journal articles, are comprehensively explored with particular attention given to applications of regularization methods as well as to practical methods used in computational analysis.

During the 90s robust control theory has seen major advances and achieved a new maturity, centered around the notion of convexity. The goal of this book is to give a graduate-level course on this theory that emphasizes these new developments, but at the same time conveys the main principles and ubiquitous tools at the heart of the subject. Its pedagogical objectives are to introduce a coherent and unified framework for studying the theory, to provide students with the control-theoretic background required to read and contribute to the research literature, and to present the main ideas and demonstrations of the major results. The book will be of value to mathematical researchers and computer scientists, graduate students planning to do research in the area, and engineering practitioners requiring advanced control techniques.

For a long time the techniques of solving linear optimization (LP) problems improved only marginally. Fifteen years ago, however, a revolutionary discovery changed everything. A new `golden age' for optimization started, which is continuing up to the current time. What is the cause of the excitement? Techniques of linear programming formed previously an isolated body of knowledge. Then suddenly a tunnel was built linking it with a rich and promising land, part of which was already cultivated, part of which was completely unexplored. These revolutionary new techniques are now applied to solve conic linear problems. This makes it possible to model and solve large classes of essentially nonlinear optimization problems as efficiently as LP problems. This volume gives an overview of the latest developments of such `High Performance Optimization Techniques'. The first part is a thorough treatment of interior point methods for semidefinite programming problems. The second part reviews today's most exciting research topics and results in the area of convex optimization. Audience: This volume is for graduate students and researchers who are interested in modern optimization techniques.

This volume summarizes and synthesizes an aspect of research work that has been done in the area of Generalized Convexity over the past few decades. Specifically, the book focuses on V-invex functions in vector optimization that have grown out of the work of Jeyakumar and Mond in the 1990 s. The authors integrate related research into the book and demonstrate the wide context from which the area has grown and continues to grow.

* Provides a broad overview of modeling approaches and solution methodologies for addressing inventory problems, particularly the management of high cost, low demand rate service parts found in multi-echelon settings

* The text may be used in a variety of courses for first-year graduate students or senior undergraduates, or as a reference for researchers and practitioners

* A background in stochastic processes and optimization is assumed

Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func tion. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.

In this authoritative and comprehensive volume, Claude Bardos and Andrei Fursikov have drawn together an impressive array of international contributors to present important recent results and perspectives in this area. The main subjects that appear here relate largely to mathematical aspects of the theory but some novel schemes used in applied mathematics are also presented. Various topics from control theory, including Navier-Stokes equations, are covered.

There has been much excitement over the emergence of new mathematical techniques for the analysis and control of nonlinear systems. In addition, great technological advances have bolstered the impact of analytic advances and produced many new problems and applications which are nonlinear in an essential way. This book lays out in a concise mathematical framework the tools and methods of analysis which underlie this diversity of applications.

Today, the optimization of production planning processes by means of IT and quantitative methods is a de-facto standard in the energy industry. Franch et al. inChapter1andIkenouyeinChapter2giveanintroduction, overview, and reasonsforthis. Furthermore, theenergyproblemnowisnotonlyachallenging one but also one of the most important issues in the world from the political and economical points of view. In every country, the government is faced with the problem of how to adopt the system of 'Cap and Trade. ' Especially energy consuming industries, such as steel, power, oil and chemicals, are seriously confronted with this problem. VIII Preface This is also the reason why the German Operations Research Society (GOR) and one of its working groups, held a symposium with the title "Stochastic Optimization in the Energy Industry. " During the 78th meeting of the GOR working group "Praxis der Mathematischen Optimierung/Real World Optimization" in Aachen at Procom GmbH on April 21/22, 2007, the speakers with an application background explained their requirements for stochasticoptimizationsolutionsbasedonpracticalexperiences. Thespeakers from the research side and the software system suppliers examined di?erent aspects of the whole subject - from the integration of wind energy, the chain of errors in nuclear power plants and the scheduling of hydroelectric power stations, and the risk assessment in trading activities to the various software systems which support stochastic optimization methods. The symposium o?ered an interesting overview which re?ected the - quirements, possibilities and restrictions of "Stochastic Optimization in the Energy Industry.

Industrial production is one of the most basic human activities indispensable to the economic activity. Due to its complexity, production is not very well understood and modeled as opposed to traditional fields of inquiry such as physics. This book aims at enhancing rigorous understanding of a particular area of production, that of analysis and optimization of production lines and networks using discrete event models and simulation. To our knowledge, this is the first book treating this subject from the point of view mentioned above. We have arrived at the realization that discrete event models and simulation provide perhaps the best tools to model production lines and networks for a number of reasons. Analysis is precise but demands enormous computational resources, usually unavailable in practical situations. Brute force simulation is also precise but slow when quick decisions are to be made. Approximate analytical models are fast but often unreliable as far as accuracy is concerned. The approach of the book, on the other hand, combines speed and accuracy to an exceptional degree in most practical applications.

Conjugate direction methods were proposed in the early 1950s. When high speed digital computing machines were developed, attempts were made to lay the fo- dations for the mathematical aspects of computations which could take advantage of the ef?ciency of digital computers. The National Bureau of Standards sponsored the Institute for Numerical Analysis, which was established at the University of California in Los Angeles. A seminar held there on numerical methods for linear equationswasattendedbyMagnusHestenes, EduardStiefel andCorneliusLanczos. This led to the ?rst communication between Lanczos and Hestenes (researchers of the NBS) and Stiefel (of the ETH in Zurich) on the conjugate direction algorithm. The method is attributed to Hestenes and Stiefel who published their joint paper in 1952 [101] in which they presented both the method of conjugate gradient and the conjugate direction methods including conjugate Gram-Schmidt processes. A closelyrelatedalgorithmwasproposedbyLanczos[114]whoworkedonalgorithms for determiningeigenvalues of a matrix. His iterative algorithm yields the similarity transformation of a matrix into the tridiagonal form from which eigenvalues can be well approximated.Thethree-termrecurrencerelationofthe Lanczosprocedurecan be obtained by eliminating a vector from the conjugate direction algorithm scheme. Initially the conjugate gradient algorithm was called the Hestenes-Stiefel-Lanczos method [86].

Now fully updated in a third edition, this is a comprehensive textbook on combinatorial optimization. It puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete but concise proofs, also for many deep results, some of which have not appeared in print before. Recent topics are covered as well, and numerous references are provided. This third edition contains a new chapter on facility location problems, an area which has been extremely active in the past few years. Furthermore there are several new sections and further material on various topics. New exercises and updates in the bibliography were added.

Substantially revised, reorganised and updated, the second edition now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems.

As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem.

Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject.

This volume deals with controllability and observability properties of nonlinear systems, as well as various ways to obtain input-output representations. The emphasis is on fundamental notions as (controlled) invariant distributions and submanifolds, together with algorithms to compute the required feedbacks.

This book offers a systematic presentation of up-to-date material scattered throughout the literature from the methodology point of view. It reviews the basic theories and methods, with many interesting problems in partial and ordinary differential equations, differential geometry and mathematical physics as applications, and provides the necessary preparation for almost all important aspects in contemporary studies. All methods are illustrated by carefully chosen examples from mechanics, physics, engineering and geometry.

Performance optimization is vital in the design and operation of modern engineering systems, including communications, manufacturing, robotics, and logistics. Most engineering systems are too complicated to model, or the system parameters cannot be easily identified, so learning techniques have to be applied. This book provides a unified framework based on a sensitivity point of view. It also introduces new approaches and proposes new research topics within this sensitivity-based framework. This new perspective on a popular topic is presented by a well respected expert in the field.

Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether 's theorem. The text includes numerous examples along with problems to help students consolidate the material.

"Each chapter contains a well-written introduction and notes. They include the author's deep insights on the subject matter and provide historical comments and guidance to related literature. This book may well become an important milestone in the literature of optimal control." -Mathematical Reviews "Thanks to a great effort to be self-contained, [this book] renders accessibly the subject to a wide audience. Therefore, it is recommended to all researchers and professionals interested in Optimal Control and its engineering and economic applications. It can serve as an excellent textbook for graduate courses in Optimal Control (with special emphasis on Nonsmooth Analysis)." -Automatica

It is with great pleasure that I accepted invitation of Adnan Ibrahimbegovic to write this preface, for this invitation gave me the privilege to be one of the ?rsttoreadhisbookandallowedmetoonceagainemphasizetheimportance for our discipline of solid mechanics, which is currently under considerable development, to produce the reference books suitable for students and all other researchers and engineers who wish to advance their knowledge on the subject. Thesolidmechanicshascloselyfollowedtheprogressincomputerscienceand is currently undergoing a true revolution where the numerical modelling and simulations are playing the central role. In the industrial environment, the 'virtual' (or the computing science) is present everywhere in the design and engineering procedures. I have a habit of saying that the solid mechanics has become the science of modelling and inthat respectexpanded beyondits t- ditional frontiers. Several facets of current developments have already been treated in di?erent works published within the series 'Studies in mechanics of materials and structures'; for example, modelling heterogeneous materials (Besson et al. ), fracture mechanics (Leblond), computational strategies and namely LATIN method (Ladev' eze), instability problems (NQ Son) and ve- ?cation of ?nite element method (Ladev' eze-Pelle). To these (French) books, one should also add the work of Lemaitre-Chaboche on nonlinear behavior of solid materials and of Batoz on ?nite element method.

Quadratic programming (QP) is one advanced mathematical technique that allows for the optimization of a quadratic function in several variables in the presence of linear constraints. This book presents recently developed algorithms for solving large QP problems and focuses on algorithms which are, in a sense optimal, i.e., they can solve important classes of problems at a cost proportional to the number of unknowns. For each algorithm presented, the book details its classical predecessor, describes its drawbacks, introduces modifications that improve its performance, and demonstrates these improvements through numerical experiments.

This self-contained monograph can serve as an introductory text on quadratic programming for graduate students and researchers. Additionally, since the solution of many nonlinear problems can be reduced to the solution of a sequence of QP problems, it can also be used as a convenient introduction to nonlinear programming.

From its origins in the minimization of integral functionals, the notion of variations has evolved greatly in connection with applications in optimization, equilibrium, and control. This book develops a unified framework and provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, and normal integrands.

In a typical mathematical model of a controlled distributed parameter process one usually ?nds either boundary or internal locally distributed controls to serve as the means to describe the effect of external actuators on the process at hand. H- ever, these classical controls, enteringthe modelequationsas additive terms, are not suitable to deal with a vast array of processes that can change their principal intr- sic properties due to the control actions. Important examples here include (but not limitedto)thechainreaction-typeprocessesinbiomedical, nuclear, chemicalan- nancial applications, which can changetheir (reaction)rate when certain "catalysts" are applied, and the so-called "smart materials," which can, for instance, alter their frequency response. The goal of this monograph is to address the issue of global controllability of partial differential equations in the context of multiplicative (or bilinear) c- trols, which enter the model equations as coef?cients. The mathematical models of our interest include the linear and nonlinear parabolic and hyperbolic PDE's, the Schrodi ] nger equation, and coupled hybrid nonlinear distributed parameter systems associated with the swimming phenomenon. Pullman, WA, USA Alexander Khapalov January 2010 vii Preface This monograph developed from the research conducted in 2001-2009 in the area of controllability theory of partial differential equations. The concept of control- bility is a principal component of Control Theory which was brought to life in the 1950's by numerous applications in engineering, and has received the most sign- icant attention both from the engineering and the mathematical communities since then."

Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

This book presents the mathematical foundations of systems theory in a self-contained, comprehensive, detailed and mathematically rigorous way. It is devoted to the analysis of dynamical systems and combines features of a detailed introductory textbook with that of a reference source. The book contains many examples and figures illustrating the text which help to bring out the intuitive ideas behind the mathematical constructions.