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Variable Ordering Structures In Vector Optimization
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Book Synopsis Variable Ordering Structures in Vector Optimization by : Gabriele Eichfelder
Download or read book Variable Ordering Structures in Vector Optimization written by Gabriele Eichfelder and published by Springer Science & Business Media. This book was released on 2014-04-04 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to vector optimization with variable ordering structures, i.e., to optimization problems with a vector-valued objective function where the elements in the objective space are compared based on a variable ordering structure: instead of a partial ordering defined by a convex cone, we see a whole family of convex cones, one attached to each element of the objective space. The book starts by presenting several applications that have recently sparked new interest in these optimization problems, and goes on to discuss fundamentals and important results on a wide range of topics. The theory developed includes various optimality notions, linear and nonlinear scalarization functionals, optimality conditions of Fermat and Lagrange type, existence and duality results. The book closes with a collection of numerical approaches for solving these problems in practice.
Book Synopsis Vector Optimization Problems with Variable Ordering Structures by : Behnam Soleimani
Download or read book Vector Optimization Problems with Variable Ordering Structures written by Behnam Soleimani and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Vector optimization; Variable ordering structures; Approximate solutions; Scalarization; Separation theorems; Ekeland's variational principle; Optimality conditions
Book Synopsis Optimality conditions for vector optimization problems with variable ordering structures by :
Download or read book Optimality conditions for vector optimization problems with variable ordering structures written by and published by . This book was released on 2010 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Properly Optimal Elements in Vector Optimization with Variable Ordering Structures by : Gabriele Eichfelder
Download or read book Properly Optimal Elements in Vector Optimization with Variable Ordering Structures written by Gabriele Eichfelder and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions.
Book Synopsis Ekeland's Variational Principle for Vector Optimization with Variable Ordering Structure by :
Download or read book Ekeland's Variational Principle for Vector Optimization with Variable Ordering Structure written by and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: There are many generalizations of Ekeland's variational principle for vector optimization problems with fixed ordering structures, i.e., ordering cones. These variational principles are useful for deriving optimality conditions, epsilon-Kolmogorov conditions in approximation theory, and epsilon-maximum principles in optimal control. Here, we present several generalizations of Ekeland's variational principle for vector optimization problems with respect to variable ordering structures. For deriving these variational principles we use nonlinear scalarization techniques. Furthermore, we derive necessary conditions for approximate solutions of vector optimization problems with respect to variable ordering structures using these variational principles and the subdifferential calculus by Mordukhovich.
Book Synopsis Ordering Structures in Vector Optimization and Applications in Medical Engineering by : Gabriele Eichfelder
Download or read book Ordering Structures in Vector Optimization and Applications in Medical Engineering written by Gabriele Eichfelder and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space R^m this can be done componentwise. That corresponds to the notion of an Edgeworth-Pareto-optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.
Book Synopsis Vector Optimization with a Variable Ordering Structure by : Gabriele Eichfelder
Download or read book Vector Optimization with a Variable Ordering Structure written by Gabriele Eichfelder and published by . This book was released on 2009 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Vector-valued Ekeland's Variational Principle in Vector Optimization with Variable Ordering Structures by : Behnam Soleimani
Download or read book A Vector-valued Ekeland's Variational Principle in Vector Optimization with Variable Ordering Structures written by Behnam Soleimani and published by . This book was released on 2016 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Optimality Conditions for Approximate Solutions of Vector Optimization Problems with Variable Ordering Structures by : Christiane Tammer
Download or read book Optimality Conditions for Approximate Solutions of Vector Optimization Problems with Variable Ordering Structures written by Christiane Tammer and published by . This book was released on 2014 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Characterization of Proper Optimal Elements with Variable Ordering Structures by : Gabriele Eichfelder
Download or read book Characterization of Proper Optimal Elements with Variable Ordering Structures written by Gabriele Eichfelder and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In vector optimization with a variable ordering structure the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. As these vector optimization problems are not only of interest in applications but also mathematical challenging, in recent publications it was started to develop a comprehensive theory. In doing that also notions of proper efficiency where generalized to variable ordering structures. In this paper we study the relations between several types of proper optimality notions, among others based on local and global approximations of the considered sets. We give scalarization results based on new functionals defined by elements from the dual cones which allow characterizations also in the nonconvex case.
Book Synopsis Recent Developments in Vector Optimization by : Qamrul Hasan Ansari
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.
Book Synopsis Variational Analysis and Set Optimization by : Akhtar A. Khan
Download or read book Variational Analysis and Set Optimization written by Akhtar A. Khan and published by CRC Press. This book was released on 2019-06-07 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the latest advances in variational analysis and set / vector optimization, including uncertain optimization, optimal control and bilevel optimization. Recent developments concerning scalarization techniques, necessary and sufficient optimality conditions and duality statements are given. New numerical methods for efficiently solving set optimization problems are provided. Moreover, applications in economics, finance and risk theory are discussed. Summary The objective of this book is to present advances in different areas of variational analysis and set optimization, especially uncertain optimization, optimal control and bilevel optimization. Uncertain optimization problems will be approached from both a stochastic as well as a robust point of view. This leads to different interpretations of the solutions, which widens the choices for a decision-maker given his preferences. Recent developments regarding linear and nonlinear scalarization techniques with solid and nonsolid ordering cones for solving set optimization problems are discussed in this book. These results are useful for deriving optimality conditions for set and vector optimization problems. Consequently, necessary and sufficient optimality conditions are presented within this book, both in terms of scalarization as well as generalized derivatives. Moreover, an overview of existing duality statements and new duality assertions is given. The book also addresses the field of variable domination structures in vector and set optimization. Including variable ordering cones is especially important in applications such as medical image registration with uncertainties. This book covers a wide range of applications of set optimization. These range from finance, investment, insurance, control theory, economics to risk theory. As uncertain multi-objective optimization, especially robust approaches, lead to set optimization, one main focus of this book is uncertain optimization. Important recent developments concerning numerical methods for solving set optimization problems sufficiently fast are main features of this book. These are illustrated by various examples as well as easy-to-follow-steps in order to facilitate the decision process for users. Simple techniques aimed at practitioners working in the fields of mathematical programming, finance and portfolio selection are presented. These will help in the decision-making process, as well as give an overview of nondominated solutions to choose from.
Book Synopsis Concepts of approximate solutions of vector optimization problems with variable order structure by : Behnam Soleimani
Download or read book Concepts of approximate solutions of vector optimization problems with variable order structure written by Behnam Soleimani and published by . This book was released on 2013 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Concepts for Approximate Solutions of Vector Optimization Problems with Variable Order Structure by : Christiane Tammer
Download or read book Concepts for Approximate Solutions of Vector Optimization Problems with Variable Order Structure written by Christiane Tammer and published by . This book was released on 2013 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Scalarization of Approximate Solutions of Vector Optimization Problem with Variable Order Structure Based on Nonlinear Scalarization by : Behnam Soleimani
Download or read book Scalarization of Approximate Solutions of Vector Optimization Problem with Variable Order Structure Based on Nonlinear Scalarization written by Behnam Soleimani and published by . This book was released on 2013 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Variational Analysis and Set Optimization by : Akhtar A. Khan
Download or read book Variational Analysis and Set Optimization written by Akhtar A. Khan and published by CRC Press. This book was released on 2019-06-07 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the latest advances in variational analysis and set / vector optimization, including uncertain optimization, optimal control and bilevel optimization. Recent developments concerning scalarization techniques, necessary and sufficient optimality conditions and duality statements are given. New numerical methods for efficiently solving set optimization problems are provided. Moreover, applications in economics, finance and risk theory are discussed. Summary The objective of this book is to present advances in different areas of variational analysis and set optimization, especially uncertain optimization, optimal control and bilevel optimization. Uncertain optimization problems will be approached from both a stochastic as well as a robust point of view. This leads to different interpretations of the solutions, which widens the choices for a decision-maker given his preferences. Recent developments regarding linear and nonlinear scalarization techniques with solid and nonsolid ordering cones for solving set optimization problems are discussed in this book. These results are useful for deriving optimality conditions for set and vector optimization problems. Consequently, necessary and sufficient optimality conditions are presented within this book, both in terms of scalarization as well as generalized derivatives. Moreover, an overview of existing duality statements and new duality assertions is given. The book also addresses the field of variable domination structures in vector and set optimization. Including variable ordering cones is especially important in applications such as medical image registration with uncertainties. This book covers a wide range of applications of set optimization. These range from finance, investment, insurance, control theory, economics to risk theory. As uncertain multi-objective optimization, especially robust approaches, lead to set optimization, one main focus of this book is uncertain optimization. Important recent developments concerning numerical methods for solving set optimization problems sufficiently fast are main features of this book. These are illustrated by various examples as well as easy-to-follow-steps in order to facilitate the decision process for users. Simple techniques aimed at practitioners working in the fields of mathematical programming, finance and portfolio selection are presented. These will help in the decision-making process, as well as give an overview of nondominated solutions to choose from.
Author :Ricardo H.C. Takahashi Publisher :Springer Science & Business Media ISBN 13 :3642198929 Total Pages :634 pages Book Rating :4.6/5 (421 download)
Book Synopsis Evolutionary Multi-Criterion Optimization by : Ricardo H.C. Takahashi
Download or read book Evolutionary Multi-Criterion Optimization written by Ricardo H.C. Takahashi and published by Springer Science & Business Media. This book was released on 2011-03-14 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 6th International Conference on Evolutionary Multi-Criterion Optimization, EMO 2011, held in Ouro Preto, Brazil, in April 2011. The 42 revised full papers presented were carefully reviewed and selected from 83 submissions. The papers deal with fundamental questions of EMO theory, such as the development of algorithmically efficient tools for the evaluation of solution-set quality , the theoretical questions related to solution archiving and others. They report on the continuing effort in the development of algorithms, either for dealing with particular classes of problems or for new forms of processing the problem information. Almost one third of the papers is related to EMO applications in a diversity of fields. Eleven papers are devoted to promote the interaction with the related field of Multi-Criterion Decision Making (MCDM).
