Vector Optimization
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Vector Optimization
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Author : Johannes Jahn
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-11-22
Vector Optimization written by Johannes Jahn and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-22 with Business & Economics categories.
Fundamentals and important results of vector optimization in a general setting are presented in this book. The theory developed includes scalarization, existence theorems, a generalized Lagrange multiplier rule and duality results. Applications to vector approximation, cooperative game theory and multiobjective optimization are described. The theory is extended to set optimization with particular emphasis on contingent epiderivatives, subgradients and optimality conditions. Background material of convex analysis being necessary is concisely summarized at the beginning. This second edition contains new parts on the adaptive Eichfelder-Polak method, a concrete application to magnetic resonance systems in medical engineering and additional remarks on the contribution of F.Y. Edgeworth and V. Pareto. The bibliography is updated and includes more recent important publications.
Theory Of Vector Optimization
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Author : Dinh The Luc
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Theory Of Vector Optimization written by Dinh The Luc and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Business & Economics categories.
These notes grew out of a series of lectures given by the author at the Univer sity of Budapest during 1985-1986. Additional results have been included which were obtained while the author was at the University of Erlangen-Niirnberg under a grant of the Alexander von Humboldt Foundation. Vector optimization has two main sources coming from economic equilibrium and welfare theories of Edgeworth (1881) and Pareto (1906) and from mathemat ical backgrounds of ordered spaces of Cantor (1897) and Hausdorff (1906). Later, game theory of Borel (1921) and von Neumann (1926) and production theory of Koopmans (1951) have also contributed to this area. However, only in the fifties, after the publication of Kuhn-Tucker's paper (1951) on the necessary and sufficient conditions for efficiency, and of Deubreu's paper (1954) on valuation equilibrium and Pareto optimum, has vector optimization been recognized as a mathematical discipline. The stretching development of this field began later in the seventies and eighties. Today there are a number of books on vector optimization. Most of them are concerned with the methodology and the applications. Few of them offer a systematic study of the theoretical aspects. The aim of these notes is to pro vide a unified background of vector optimization,with the emphasis on nonconvex problems in infinite dimensional spaces ordered by convex cones. The notes are arranged into six chapters. The first chapter presents prelim inary material.
Vector Optimization
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Author : Guang-ya Chen
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-11-20
Vector Optimization written by Guang-ya Chen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-20 with Business & Economics categories.
Vector optimization model has found many important applications in decision making problems such as those in economics theory, management science, and engineering design (since the introduction of the Pareto optimal solu tion in 1896). Typical examples of vector optimization model include maxi mization/minimization of the objective pairs (time, cost), (benefit, cost), and (mean, variance) etc. Many practical equilibrium problems can be formulated as variational in equality problems, rather than optimization problems, unless further assump tions are imposed. The vector variational inequality was introduced by Gi- nessi (1980). Extensive research on its relations with vector optimization, the existence of a solution and duality theory has been pursued. The fundamental idea of the Ekeland's variational principle is to assign an optimization problem a slightly perturbed one having a unique solution which is at the same time an approximate solution of the original problem. This principle has been an important tool for nonlinear analysis and optimization theory. Along with the development of vector optimization and set-valued optimization, the vector variational principle introduced by Nemeth (1980) has been an interesting topic in the last decade. Fan Ky's minimax theorems and minimax inequalities for real-valued func tions have played a key role in optimization theory, game theory and math ematical economics. An extension was proposed to vector payoffs was intro duced by Blackwell (1955).
Vector Optimization With Infimum And Supremum
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Author : Andreas Löhne
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-05-25
Vector Optimization With Infimum And Supremum written by Andreas Löhne and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-05-25 with Business & Economics categories.
The theory of Vector Optimization is developed by a systematic usage of infimum and supremum. In order to get existence and appropriate properties of the infimum, the image space of the vector optimization problem is embedded into a larger space, which is a subset of the power set, in fact, the space of self-infimal sets. Based on this idea we establish solution concepts, existence and duality results and algorithms for the linear case. The main advantage of this approach is the high degree of analogy to corresponding results of Scalar Optimization. The concepts and results are used to explain and to improve practically relevant algorithms for linear vector optimization problems.
Nonsmooth Vector Functions And Continuous Optimization
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Author : V. Jeyakumar
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-10-23
Nonsmooth Vector Functions And Continuous Optimization written by V. Jeyakumar and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-10-23 with Mathematics categories.
Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems, as well as variational inequalities in finite dimensions. The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus, using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function.
Theory And Methods Of Vector Optimization Volume One
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Author : Yu. K. Mashunin
language : en
Publisher: Cambridge Scholars Publishing
Release Date : 2020-03-24
Theory And Methods Of Vector Optimization Volume One written by Yu. K. Mashunin and has been published by Cambridge Scholars Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-24 with Mathematics categories.
This first volume presents the theory and methods of solving vector optimization problems, using initial definitions that include axioms and the optimality principle. This book proves, mathematically, that the result it presents for the solution of the vector (multi-criteria) problem is the optimal outcome, and, as such, solves the problem of vector optimization for the first time. It shows that applied methods of solving vector optimization problems can be used by researchers in modeling and simulating the development of economic systems and technical (engineering) systems.
Optimization By Vector Space Methods
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Author : David G. Luenberger
language : en
Publisher: John Wiley & Sons
Release Date : 1997-01-23
Optimization By Vector Space Methods written by David G. Luenberger and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-23 with Technology & Engineering categories.
Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.
Regularization Optimization Kernels And Support Vector Machines
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Author : Johan A.K. Suykens
language : en
Publisher: CRC Press
Release Date : 2014-10-23
Regularization Optimization Kernels And Support Vector Machines written by Johan A.K. Suykens and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-23 with Computers categories.
Regularization, Optimization, Kernels, and Support Vector Machines offers a snapshot of the current state of the art of large-scale machine learning, providing a single multidisciplinary source for the latest research and advances in regularization, sparsity, compressed sensing, convex and large-scale optimization, kernel methods, and support vector machines. Consisting of 21 chapters authored by leading researchers in machine learning, this comprehensive reference: Covers the relationship between support vector machines (SVMs) and the Lasso Discusses multi-layer SVMs Explores nonparametric feature selection, basis pursuit methods, and robust compressive sensing Describes graph-based regularization methods for single- and multi-task learning Considers regularized methods for dictionary learning and portfolio selection Addresses non-negative matrix factorization Examines low-rank matrix and tensor-based models Presents advanced kernel methods for batch and online machine learning, system identification, domain adaptation, and image processing Tackles large-scale algorithms including conditional gradient methods, (non-convex) proximal techniques, and stochastic gradient descent Regularization, Optimization, Kernels, and Support Vector Machines is ideal for researchers in machine learning, pattern recognition, data mining, signal processing, statistical learning, and related areas.
Recent Advances And Historical Development Of Vector Optimization
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Author : Johannes Jahn
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Recent Advances And Historical Development Of Vector Optimization written by Johannes Jahn and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Business & Economics categories.
In vector optimization one investigates optimization problems in an abstract setting which have a not necessarily real-valued objective function. This scientific discipline is closely related to multi-objective optimization and multi-criteria decision making. This book contains refereed contributions to the "International Conference on Vector Optimization" held at the Technical University of Darmstadt from August 4-7, 1986. This meeting was an interdisciplinary forum devoted to new results in the theory, to applications as well as to the solution of vector optimization problems which are relevant in practice. Because of the great variety of topics covered by the contributions, the 25 articles of this volume are organized in different sections: Historical retrospect, mathematical theory, goal setting and decision making, engineering applications, and related topics. The papers of the invited State-of-the-Art Tutorials given by Professors J.M. Borwein, H. Eschenauer, W. Stadler and P.L. Yu are also included.
Convex Optimization
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Author : Stephen P. Boyd
language : en
Publisher: Cambridge University Press
Release Date : 2004-03-08
Convex Optimization written by Stephen P. Boyd and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-03-08 with Business & Economics categories.
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.