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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.
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.
Like norms, translation invariant functions are a natural and
powerful tool for the separation of sets and scalarization. This
book provides an extensive foundation for their application. It
presents in a unified way new results as well as results which are
scattered throughout the literature. The functions are defined on
linear spaces and can be applied to nonconvex problems. Fundamental
theorems for the function class are proved, with implications for
arbitrary extended real-valued functions. The scope of applications
is illustrated by chapters related to vector optimization,
set-valued optimization, and optimization under uncertainty, by
fundamental statements in nonlinear functional analysis and by
examples from mathematical finance as well as from consumer and
production theory. The book is written for students and researchers
in mathematics and mathematical economics. Engineers and
researchers from other disciplines can benefit from the
applications, for example from scalarization methods for
multiobjective optimization and optimal control problems.
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.
In mathematical modeling of processes one often encounters optimization problems involving more than one objective function, so that Multiobjective Optimization (or Vector Optimization) has received new impetus. The growing interest in multiobjective problems, both from the theoretical point of view and as it concerns applications to real problems, asks for a general scheme which embraces several existing developments and stimulates new ones. In this book the authors provide the newest results and applications of this quickly growing field. This book will be of interest to graduate students in mathematics, economics, and engineering, as well as researchers in pure and applied mathematics, economics, engineering, geography, and town planning. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book.
This book discusses basic tools of partially ordered spaces and
applies them to variational methods in Nonlinear Analysis and for
optimizing problems. This book is aimed at graduate students and
research mathematicians.
In diesem Lehrbuch werden die für die Wirtschaftsmathematik,
insbesondere für die Optimierungstheorie, Stochastik und Numerik,
erforderlichen Grundlagen der Funktionalanalysis in einer
anschaulichen Form mit Bezügen zu den entsprechenden Anwendungen
in jedem Kapitel dargestellt. Dabei wird eine Untergliederung
entsprechend der für die Wirtschaftsmathematik relevanten
Hauptsätze der Funktionalanalysis, wie Baire's Kategoriesatz,
Approximations- und Projektionssatz, Hahn-Banach-Theorem,
Fixpunktaussagen und KKM-Theorem und Variationsprinzipien,
vorgenommen.
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.
In diesem Buch wird die Vielgestaltigkeit von Optimierung und
Approximation zusammen mit ihrem breiten Umfeld anhand von Aufgaben
samt ihren Loesungen und nutzlichen Anwendungen zum Ausdruck
gebracht. Fachlich steht dabei im Vordergrund, Methoden der
Angewandten Analysis zu nutzen, um die Struktur und Eigenschaften
der Probleme zu erkennen und handhabbare Optimalitatsbedingungen
herzuleiten, die die Behandlung der Aufgaben ermoeglichen und
vereinfachen. Viele praktische Aufgabenstellungen fuhren auf
konvexe bzw. nichtkonvexe Optimierungsprobleme, Mehrkriterielle
Optimierungsprobleme, Standortprobleme, Probleme der Risikotheorie,
Versicherungsmathematik, Optimierungsprobleme mit Unsicherheiten
und Modelle aus der Signaltheorie, die in den behandelten Aufgaben
diskutiert werden. Hinweise auf online verfugbare Software werden
gegeben. Das Buch richtet sich an Studierende und Lehrende der
Mathematik, Wirtschaftsmathematik, Informatik, Physik, den
Wirtschafts- und Ingenieurwissenschaften (u.a. der Mechatronik).
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