Ekeland S Variational Principle For Vector Optimization With Variable Ordering Structure
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Ekeland S Variational Principle For Vector Optimization With Variable Ordering Structure
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Author :
language : en
Publisher:
Release Date : 2014
Ekeland S Variational Principle For Vector Optimization With Variable Ordering Structure written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
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.
Variable Ordering Structures In Vector Optimization
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Author : Gabriele Eichfelder
language : en
Publisher: Springer Science & Business Media
Release Date : 2014-04-04
Variable Ordering Structures In Vector Optimization written by Gabriele Eichfelder 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 2014-04-04 with Mathematics categories.
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.
Optimization Variational Analysis And Applications
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Author : Vivek Laha
language : en
Publisher: Springer Nature
Release Date : 2021-07-27
Optimization Variational Analysis And Applications written by Vivek Laha and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-27 with Mathematics categories.
This book includes selected papers presented at the Indo-French Seminar on Optimization, Variational Analysis and Applications (IFSOVAA-2020), held at the Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India, from 2–4 February 2020. The book discusses current optimization problems and their solutions by using the powerful tool of variational analysis. Topics covered in this volume include set optimization, multiobjective optimization, mathematical programs with complementary, equilibrium, vanishing and switching constraints, copositive optimization, interval-valued optimization, sequential quadratic programming, bound-constrained optimization, variational inequalities, and more. Several applications in different branches of applied mathematics, engineering, economics, finance, and medical sciences have been included. Each chapter not only provides a detailed survey of the topic but also builds systematic theories and suitable algorithms to deduce the most recent findings in literature. This volume appeals to graduate students as well as researchers and practitioners in pure and applied mathematics and related fields that make use of variational analysis in solving optimization problems.
Convex Optimization Theory Algorithms And Applications
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Author : Balendu Bhooshan Upadhyay
language : en
Publisher: Springer Nature
Release Date : 2025-04-25
Convex Optimization Theory Algorithms And Applications written by Balendu Bhooshan Upadhyay and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-25 with Mathematics categories.
This volume includes chapters on topics presented at the conference on Recent Trends in Convex Optimization: Theory, Algorithms and Applications (RTCOTAA-2020), held at the Department of Mathematics, Indian Institute of Technology Patna, Bihar, India, from 29–31 October 2020. It discusses a comprehensive exploration of the realm of optimization, encompassing both the theoretical underpinnings and the multifaceted real-life implementations of the optimization theory. It meticulously features essential optimization concepts, such as convex analysis, generalized convexity, monotonicity, etc., elucidating their theoretical advancements and significance in the optimization sphere. Multiobjective optimization is a pivotal topic which addresses the inherent difficulties faced in conflicting objectives. The book delves into various theoretical concepts and covers some practical algorithmic approaches to solve multiobjective optimization, such as the line search and the enhanced non-monotone quasi-Newton algorithms. It also deliberates on several other significant topics in optimization, such as the perturbation approach for vector optimization, and solution methods for set-valued optimization. Nonsmooth optimization is extensively covered, with in-depth discussions on various well-known tools of nonsmooth analysis, such as convexificators, limiting subdifferentials, tangential subdifferentials, quasi-differentials, etc. Notable optimization algorithms, such as the interior point algorithm and Lemke’s algorithm, are dissected in detail, offering insights into their applicability and effectiveness. The book explores modern applications of optimization theory, for instance, optimized image encryption, resource allocation, target tracking problems, deep learning, entropy optimization, etc. Ranging from gradient-based optimization algorithms to metaheuristic approaches such as particle swarm optimization, the book navigates through the intersection of optimization theory and deep learning, thereby unravelling new research perspectives in artificial intelligence, machine learning and other fields of modern science. Designed primarily for graduate students and researchers across a variety of disciplines such as mathematics, operations research, electrical and electronics engineering, computer science, robotics, deep learning, image processing and artificial intelligence, this book serves as a comprehensive resource for someone interested in exploring the multifaceted domain of mathematical optimization and its myriad applications.
Variational Methods In Partially Ordered Spaces
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Author : Alfred Göpfert
language : en
Publisher: Springer Nature
Release Date : 2023-12-08
Variational Methods In Partially Ordered Spaces written by Alfred Göpfert and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-08 with Business & Economics categories.
In mathematical modeling of processes occurring in logistics, management science, operations research, networks, mathematical finance, medicine, and control theory, one often encounters optimization problems involving more than one objective function so that Multiobjective Optimization (or Vector Optimization, initiated by W. Pareto) has received new impetus. The growing interest in vector optimization problems, both from the theoretical point of view and as it concerns applications to real world optimization problems, asks for a general scheme which embraces several existing developments and stimulates new ones. This book aims to provide the newest results and applications of this quickly growing field. Basic tools of partially ordered spaces are discussed and applied to variational methods in nonlinear analysis and to optimization problems. The book begins by providing simple examples that illustrate what kind of problems can be handled with the methods presented. The book then deals with connections between order structures and topological structures of sets, discusses properties of nonlinear scalarization functions, and derives corresponding separation theorems for not necessarily convex sets. Furthermore, characterizations of set relations via scalarization are presented. Important topological properties of multifunctions and new results concerning the theory of vector optimization and equilibrium problems are presented in the book. These results are applied to construct numerical algorithms, especially, proximal-point algorithms and geometric algorithms based on duality assertions. In the second edition, new sections about set less relations, optimality conditions in set optimization and the asymptotic behavior of multiobjective Pareto-equilibrium problems have been incorporated. Furthermore, a new chapter regarding scalar optimization problems under uncertainty and robust counterpart problems employing approaches based on vector optimization, set optimization, and nonlinear scalarization was added. Throughout the entire book, there are examples used to illustrate the results and check the stated conditions. This book will be of interest to graduate students and researchers in pure and applied mathematics, economics, and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book.
Modelling Computation And Optimization In Information Systems And Management Sciences
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Author : Hoai An Le Thi
language : en
Publisher: Springer
Release Date : 2015-05-04
Modelling Computation And Optimization In Information Systems And Management Sciences written by Hoai An Le Thi and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-04 with Technology & Engineering categories.
This proceedings set contains 85 selected full papers presented at the 3rd International Conference on Modelling, Computation and Optimization in Information Systems and Management Sciences - MCO 2015, held on May 11–13, 2015 at Lorraine University, France. The present part I of the 2 volume set includes articles devoted to Combinatorial optimization and applications, DC programming and DCA: thirty years of Developments, Dynamic Optimization, Modelling and Optimization in financial engineering, Multiobjective programming, Numerical Optimization, Spline Approximation and Optimization, as well as Variational Principles and Applications.
Vector Variational Inequalities And Vector Optimization
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Author : Qamrul Hasan Ansari
language : en
Publisher: Springer
Release Date : 2017-10-31
Vector Variational Inequalities And Vector Optimization written by Qamrul Hasan Ansari and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-31 with Business & Economics categories.
This book presents the mathematical theory of vector variational inequalities and their relations with vector optimization problems. It is the first-ever book to introduce well-posedness and sensitivity analysis for vector equilibrium problems. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, set-valued analysis and fixed-point theory for set-valued maps, as well as a brief introduction to variational inequalities and equilibrium problems. Chapter 2 presents an overview of analysis over cones, including continuity and convexity of vector-valued functions. The book then shifts its focus to solution concepts and classical methods in vector optimization. It describes the formulation of vector variational inequalities and their applications to vector optimization, followed by separate chapters on linear scalarization, nonsmooth and generalized vector variational inequalities. Lastly, the book introduces readers to vector equilibrium problems and generalized vector equilibrium problems. Written in an illustrative and reader-friendly way, the book offers a valuable resource for all researchers whose work involves optimization and vector optimization.
Variational Analysis And Applications
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Author : Boris S. Mordukhovich
language : en
Publisher: Springer
Release Date : 2018-08-02
Variational Analysis And Applications written by Boris S. Mordukhovich and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-02 with Mathematics categories.
Building on fundamental results in variational analysis, this monograph presents new and recent developments in the field as well as selected applications. Accessible to a broad spectrum of potential readers, the main material is presented in finite-dimensional spaces. Infinite-dimensional developments are discussed at the end of each chapter with comprehensive commentaries which emphasize the essence of major results, track the genesis of ideas, provide historical comments, and illuminate challenging open questions and directions for future research. The first half of the book (Chapters 1–6) gives a systematic exposition of key concepts and facts, containing basic material as well as some recent and new developments. These first chapters are particularly accessible to masters/doctoral students taking courses in modern optimization, variational analysis, applied analysis, variational inequalities, and variational methods. The reader’s development of skills will be facilitated as they work through each, or a portion of, the multitude of exercises of varying levels. Additionally, the reader may find hints and references to more difficult exercises and are encouraged to receive further inspiration from the gems in chapter commentaries. Chapters 7–10 focus on recent results and applications of variational analysis to advanced problems in modern optimization theory, including its hierarchical and multiobjective aspects, as well as microeconomics, and related areas. It will be of great use to researchers and professionals in applied and behavioral sciences and engineering.
Applications Of Nonlinear Analysis
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Author : Themistocles M. Rassias
language : en
Publisher: Springer
Release Date : 2018-06-29
Applications Of Nonlinear Analysis written by Themistocles M. Rassias and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-29 with Mathematics categories.
New applications, research, and fundamental theories in nonlinear analysis are presented in this book. Each chapter provides a unique insight into a large domain of research focusing on functional equations, stability theory, approximation theory, inequalities, nonlinear functional analysis, and calculus of variations with applications to optimization theory. Topics include: Fixed point theory Fixed-circle theory Coupled fixed points Nonlinear duality in Banach spaces Jensen's integral inequality and applications Nonlinear differential equations Nonlinear integro-differential equations Quasiconvexity, Stability of a Cauchy-Jensen additive mapping Generalizations of metric spaces Hilbert-type integral inequality, Solitons Quadratic functional equations in fuzzy Banach spaces Asymptotic orbits in Hill’sproblem Time-domain electromagnetics Inertial Mann algorithms Mathematical modelling Robotics Graduate students and researchers will find this book helpful in comprehending current applications and developments in mathematical analysis. Research scientists and engineers studying essential modern methods and techniques to solve a variety of problems will find this book a valuable source filled with examples that illustrate concepts.
Variational Analysis And Set Optimization
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Author : Akhtar A. Khan
language : en
Publisher: CRC Press
Release Date : 2019-06-07
Variational Analysis And Set Optimization written by Akhtar A. Khan and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-07 with Business & Economics categories.
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.