Download or read book Well Posed Optimization Problems written by Assen L. Dontchev and published by Springer. This book was released on 2006-11-15 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in a unified way the mathematical theory of well-posedness in optimization. The basic concepts of well-posedness and the links among them are studied, in particular Hadamard and Tykhonov well-posedness. Abstract optimization problems as well as applications to optimal control, calculus of variations and mathematical programming are considered. Both the pure and applied side of these topics are presented. The main subject is often introduced by heuristics, particular cases and examples. Complete proofs are provided. The expected knowledge of the reader does not extend beyond textbook (real and functional) analysis, some topology and differential equations and basic optimization. References are provided for more advanced topics. The book is addressed to mathematicians interested in optimization and related topics, and also to engineers, control theorists, economists and applied scientists who can find here a mathematical justification of practical procedures they encounter.
Download or read book Well Posed Optimization Problems written by Asen L. Dontchev and published by Springer. This book was released on 2014-03-12 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in a unified way the mathematical theory of well-posedness in optimization. The basic concepts of well-posedness and the links among them are studied, in particular Hadamard and Tykhonov well-posedness. Abstract optimization problems as well as applications to optimal control, calculus of variations and mathematical programming are considered. Both the pure and applied side of these topics are presented. The main subject is often introduced by heuristics, particular cases and examples. Complete proofs are provided. The expected knowledge of the reader does not extend beyond textbook (real and functional) analysis, some topology and differential equations and basic optimization. References are provided for more advanced topics. The book is addressed to mathematicians interested in optimization and related topics, and also to engineers, control theorists, economists and applied scientists who can find here a mathematical justification of practical procedures they encounter.
Download or read book Convexity and Well Posed Problems written by Roberto Lucchetti and published by Springer Science & Business Media. This book was released on 2005-11-16 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e. , functions taking possibly the values?? and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value ?? is allowed because useful operations, such as the inf-convolution, can give rise to functions valued?? even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives.
Download or read book Ill Posed Problems Theory and Applications written by A. Bakushinsky and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent years have been characterized by the increasing amountofpublications in the field ofso-called ill-posed problems. This is easilyunderstandable because we observe the rapid progress of a relatively young branch ofmathematics, ofwhich the first results date back to about 30 years ago. By now, impressive results have been achieved both in the theory ofsolving ill-posed problems and in the applicationsofalgorithms using modem computers. To mention just one field, one can name the computer tomography which could not possibly have been developed without modem tools for solving ill-posed problems. When writing this book, the authors tried to define the place and role of ill posed problems in modem mathematics. In a few words, we define the theory of ill-posed problems as the theory of approximating functions with approximately given arguments in functional spaces. The difference between well-posed and ill posed problems is concerned with the fact that the latter are associated with discontinuous functions. This approach is followed by the authors throughout the whole book. We hope that the theoretical results will be of interest to researchers working in approximation theory and functional analysis. As for particular algorithms for solving ill-posed problems, the authors paid general attention to the principles ofconstructing such algorithms as the methods for approximating discontinuous functions with approximately specified arguments. In this way it proved possible to define the limits of applicability of regularization techniques.
Download or read book Well posed Ill posed and Intermediate Problems with Applications written by Petrov Yuri P. and published by Walter de Gruyter. This book was released on 2011-12-22 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with one of the key problems in applied mathematics, namely the investigation into and providing for solution stability in solving equations with due allowance for inaccuracies in set initial data, parameters and coefficients of a mathematical model for an object under study, instrumental function, initial conditions, etc., and also with allowance for miscalculations, including roundoff errors. Until recently, all problems in mathematics, physics and engineering were divided into two classes: well-posed problems and ill-posed problems. The authors introduce a third class of problems: intermediate ones, which are problems that change their property of being well- or ill-posed on equivalent transformations of governing equations, and also problems that display the property of being either well- or ill-posed depending on the type of the functional space used. The book is divided into two parts: Part one deals with general properties of all three classes of mathematical, physical and engineering problems with approaches to solve them; Part two deals with several stable models for solving inverse ill-posed problems, illustrated with numerical examples.
Download or read book Recent Developments in Well Posed Variational Problems written by Roberto Lucchetti and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 271 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains several surveys focused on the ideas of approximate solutions, well-posedness and stability of problems in scalar and vector optimization, game theory and calculus of variations. These concepts are of particular interest in many fields of mathematics. The idea of stability goes back at least to J. Hadamard who introduced it in the setting of differential equations; the concept of well-posedness for minimum problems is more recent (the mid-sixties) and originates with A.N. Tykhonov. It turns out that there are connections between the two properties in the sense that a well-posed problem which, at least in principle, is "easy to solve", has a solution set that does not vary too much under perturbation of the data of the problem, i.e. it is "stable". These themes have been studied in depth for minimum problems and now we have a general picture of the related phenomena in this case. But, of course, the same concepts can be studied in other more complicated situations as, e.g. vector optimization, game theory and variational inequalities. Let us mention that in several of these new areas there is not even a unique idea of what should be called approximate solution, and the latter is at the basis of the definition of well posed problem.
Download or read book Well Posed Nonlinear Problems written by Mircea Sofonea and published by Springer Nature. This book was released on 2023-11-28 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents an original method to unify the mathematical theories of well-posed problems and contact mechanics. The author uses a new concept called the Tykhonov triple to develop a well-posedness theory in which every convergence result can be interpreted as a well-posedness result. This will be useful for studying a wide class of nonlinear problems, including fixed-point problems, inequality problems, and optimal control problems. Another unique feature of the manuscript is the unitary treatment of mathematical models of contact, for which new variational formulations and convergence results are presented. Well-Posed Nonlinear Problems will be a valuable resource for PhD students and researchers studying contact problems. It will also be accessible to interested researchers in related fields, such as physics, mechanics, engineering, and operations research.
Download or read book Separable Optimization written by Stefan M. Stefanov and published by Springer Nature. This book was released on 2022-01-01 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, the theory, methods and applications of separable optimization are considered. Some general results are presented, techniques of approximating the separable problem by linear programming problem, and dynamic programming are also studied. Convex separable programs subject to inequality/ equality constraint(s) and bounds on variables are also studied and convergent iterative algorithms of polynomial complexity are proposed. As an application, these algorithms are used in the implementation of stochastic quasigradient methods to some separable stochastic programs. The problems of numerical approximation of tabulated functions and numerical solution of overdetermined systems of linear algebraic equations and some systems of nonlinear equations are solved by separable convex unconstrained minimization problems. Some properties of the Knapsack polytope are also studied. This second edition includes a substantial amount of new and revised content. Three new chapters, 15-17, are included. Chapters 15-16 are devoted to the further analysis of the Knapsack problem. Chapter 17 is focused on the analysis of a nonlinear transportation problem. Three new Appendices (E-G) are also added to this edition and present technical details that help round out the coverage. Optimization problems and methods for solving the problems considered are interesting not only from the viewpoint of optimization theory, optimization methods and their applications, but also from the viewpoint of other fields of science, especially the artificial intelligence and machine learning fields within computer science. This book is intended for the researcher, practitioner, or engineer who is interested in the detailed treatment of separable programming and wants to take advantage of the latest theoretical and algorithmic results. It may also be used as a textbook for a special topics course or as a supplementary textbook for graduate courses on nonlinear and convex optimization.
Download or read book Counterexamples in Optimal Control Theory written by Semen Ya. Serovaiskii and published by Walter de Gruyter. This book was released on 2011-12-01 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph deals with cases where optimal control either does not exist or is not unique, cases where optimality conditions are insufficient of degenerate, or where extremum problems in the sense of Tikhonov and Hadamard are ill-posed, and other situations. A formal application of classical optimisation methods in such cases either leads to wrong results or has no effect. The detailed analysis of these examples should provide a better understanding of the modern theory of optimal control and the practical difficulties of solving extremum problems.
Download or read book Well Posed Optimization Problems written by Asen L. Dontchev and published by Springer. This book was released on 1993-06-14 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in a unified way the mathematical theory of well-posedness in optimization. The basic concepts of well-posedness and the links among them are studied, in particular Hadamard and Tykhonov well-posedness. Abstract optimization problems as well as applications to optimal control, calculus of variations and mathematical programming are considered. Both the pure and applied side of these topics are presented. The main subject is often introduced by heuristics, particular cases and examples. Complete proofs are provided. The expected knowledge of the reader does not extend beyond textbook (real and functional) analysis, some topology and differential equations and basic optimization. References are provided for more advanced topics. The book is addressed to mathematicians interested in optimization and related topics, and also to engineers, control theorists, economists and applied scientists who can find here a mathematical justification of practical procedures they encounter.
Download or read book Set valued Optimization written by Akhtar A. Khan and published by Springer. This book was released on 2014-10-20 with total page 781 pages. Available in PDF, EPUB and Kindle. Book excerpt: Set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objective map and/or the constraints maps are set-valued maps acting between certain spaces. Since set-valued maps subsumes single valued maps, set-valued optimization provides an important extension and unification of the scalar as well as the vector optimization problems. Therefore this relatively new discipline has justifiably attracted a great deal of attention in recent years. This book presents, in a unified framework, basic properties on ordering relations, solution concepts for set-valued optimization problems, a detailed description of convex set-valued maps, most recent developments in separation theorems, scalarization techniques, variational principles, tangent cones of first and higher order, sub-differential of set-valued maps, generalized derivatives of set-valued maps, sensitivity analysis, optimality conditions, duality and applications in economics among other things.
Download or read book Equilibrium Problems Nonsmooth Optimization and Variational Inequality Models written by F. Giannessi and published by Springer Science & Business Media. This book was released on 2006-04-11 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the book is to cover the three fundamental aspects of research in equilibrium problems: the statement problem and its formulation using mainly variational methods, its theoretical solution by means of classical and new variational tools, the calculus of solutions and applications in concrete cases. The book shows how many equilibrium problems follow a general law (the so-called user equilibrium condition). Such law allows us to express the problem in terms of variational inequalities. Variational inequalities provide a powerful methodology, by which existence and calculation of the solution can be obtained.
Download or read book Regularization Algorithms for Ill Posed Problems written by Anatoly B. Bakushinsky and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-02-05 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: This specialized and authoritative book contains an overview of modern approaches to constructing approximations to solutions of ill-posed operator equations, both linear and nonlinear. These approximation schemes form a basis for implementable numerical algorithms for the stable solution of operator equations arising in contemporary mathematical modeling, and in particular when solving inverse problems of mathematical physics. The book presents in detail stable solution methods for ill-posed problems using the methodology of iterative regularization of classical iterative schemes and the techniques of finite dimensional and finite difference approximations of the problems under study. Special attention is paid to ill-posed Cauchy problems for linear operator differential equations and to ill-posed variational inequalities and optimization problems. The readers are expected to have basic knowledge in functional analysis and differential equations. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems, and also to advanced students in these fields. Contents Introduction Regularization Methods For Linear Equations Finite Difference Methods Iterative Regularization Methods Finite-Dimensional Iterative Processes Variational Inequalities and Optimization Problems
Download or read book Recent Developments in Well Posed Variational Problems written by Roberto Lucchetti and published by Springer. This book was released on 2013-01-11 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains several surveys focused on the ideas of approximate solutions, well-posedness and stability of problems in scalar and vector optimization, game theory and calculus of variations. These concepts are of particular interest in many fields of mathematics. The idea of stability goes back at least to J. Hadamard who introduced it in the setting of differential equations; the concept of well-posedness for minimum problems is more recent (the mid-sixties) and originates with A.N. Tykhonov. It turns out that there are connections between the two properties in the sense that a well-posed problem which, at least in principle, is "easy to solve", has a solution set that does not vary too much under perturbation of the data of the problem, i.e. it is "stable". These themes have been studied in depth for minimum problems and now we have a general picture of the related phenomena in this case. But, of course, the same concepts can be studied in other more complicated situations as, e.g. vector optimization, game theory and variational inequalities. Let us mention that in several of these new areas there is not even a unique idea of what should be called approximate solution, and the latter is at the basis of the definition of well posed problem.
Download or read book Numerical Methods for the Solution of Ill Posed Problems written by A.N. Tikhonov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in science, technology and engineering are posed in the form of operator equations of the first kind, with the operator and RHS approximately known. But such problems often turn out to be ill-posed, having no solution, or a non-unique solution, and/or an unstable solution. Non-existence and non-uniqueness can usually be overcome by settling for `generalised' solutions, leading to the need to develop regularising algorithms. The theory of ill-posed problems has advanced greatly since A. N. Tikhonov laid its foundations, the Russian original of this book (1990) rapidly becoming a classical monograph on the topic. The present edition has been completely updated to consider linear ill-posed problems with or without a priori constraints (non-negativity, monotonicity, convexity, etc.). Besides the theoretical material, the book also contains a FORTRAN program library. Audience: Postgraduate students of physics, mathematics, chemistry, economics, engineering. Engineers and scientists interested in data processing and the theory of ill-posed problems.
Download or read book Recent Developments in Vector Optimization written by Qamrul Hasan Ansari and published by Springer Science & Business Media. This book was released on 2011-09-21 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt: We always come cross several decision-making problems in our daily life. Such problems are always conflicting in which many different view points should be satisfied. In politics, business, industrial systems, management science, networks, etc. one often encounters such kind of problems. The most important and difficult part in such problems is the conflict between various objectives and goals. In these problems, one has to find the minimum(or maximum) for several objective functions. Such problems are called vector optimization problems (VOP),multi-criteria optimization problems or multi-objective optimization problems. This volume deals with several different topics / aspects of vector optimization theory ranging from the very beginning to the most recent one. It contains fourteen chapters written by different experts in the field of vector optimization.
Download or read book Optimization written by Simon Serovajsky and published by CRC Press. This book was released on 2024-07-30 with total page 657 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization: 100 Examples is a book devoted to the analysis of scenarios for which the use of well-known optimization methods encounter certain difficulties. Analysing such examples allows a deeper understanding of the features of these optimization methods, including the limits of their applicability. In this way, the book seeks to stimulate further development and understanding of the theory of optimal control. The study of the presented examples makes it possible to more effectively diagnose problems that arise in the practical solution of optimal control problems, and to find ways to overcome the difficulties that have arisen. Features Vast collection of examples Simple. accessible presentation Suitable as a research reference for anyone with an interest in optimization and optimal control theory, including mathematicians and engineers Examples differ in properties, i.e. each effect for each class of problems is illustrated by a unique example. Simon Serovajsky is a professor of mathematics at Al-Farabi Kazakh National University in Kazakhstan. He is the author of many books published in the area of optimization and optimal control theory, mathematical physics, mathematical modelling, philosophy and history of mathematics as well as a long list of high-quality publications in learned journals.
