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2019 marked the 85th anniversary of Heinrich Freiherr von
Stackelberg's habilitation thesis "Marktform und Gleichgewicht,"
which formed the roots of bilevel optimization. Research on the
topic has grown tremendously since its introduction in the field of
mathematical optimization. Besides the substantial advances that
have been made from the perspective of game theory, many sub-fields
of bilevel optimization have emerged concerning optimal control,
multiobjective optimization, energy and electricity markets,
management science, security and many more. Each chapter of this
book covers a specific aspect of bilevel optimization that has
grown significantly or holds great potential to grow, and was
written by top experts in the corresponding area. In other words,
unlike other works on the subject, this book consists of surveys of
different topics on bilevel optimization. Hence, it can serve as a
point of departure for students and researchers beginning their
research journey or pursuing related projects. It also provides a
unique opportunity for experienced researchers in the field to
learn about the progress made so far and directions that warrant
further investigation. All chapters have been peer-reviewed by
experts on mathematical optimization.
2019 marked the 85th anniversary of Heinrich Freiherr von
Stackelberg's habilitation thesis "Marktform und Gleichgewicht,"
which formed the roots of bilevel optimization. Research on the
topic has grown tremendously since its introduction in the field of
mathematical optimization. Besides the substantial advances that
have been made from the perspective of game theory, many sub-fields
of bilevel optimization have emerged concerning optimal control,
multiobjective optimization, energy and electricity markets,
management science, security and many more. Each chapter of this
book covers a specific aspect of bilevel optimization that has
grown significantly or holds great potential to grow, and was
written by top experts in the corresponding area. In other words,
unlike other works on the subject, this book consists of surveys of
different topics on bilevel optimization. Hence, it can serve as a
point of departure for students and researchers beginning their
research journey or pursuing related projects. It also provides a
unique opportunity for experienced researchers in the field to
learn about the progress made so far and directions that warrant
further investigation. All chapters have been peer-reviewed by
experts on mathematical optimization.
This book describes recent theoretical findings relevant to bilevel
programming in general, and in mixed-integer bilevel programming in
particular. It describes recent applications in energy problems,
such as the stochastic bilevel optimization approaches used in the
natural gas industry. New algorithms for solving linear and
mixed-integer bilevel programming problems are presented and
explained.
This book describes recent theoretical findings relevant to bilevel
programming in general, and in mixed-integer bilevel programming in
particular. It describes recent applications in energy problems,
such as the stochastic bilevel optimization approaches used in the
natural gas industry. New algorithms for solving linear and
mixed-integer bilevel programming problems are presented and
explained.
In the field of nondifferentiable nonconvex optimization, one of
the most intensely investigated areas is that of optimization
problems involving multivalued mappings in constraints or as the
objective function. This book focuses on the tremendous development
in the field that has taken place since the publication of the most
recent volumes on the subject. The new topics studied include the
formulation of optimality conditions using different kinds of
generalized derivatives for set-valued mappings (such as, for
example, the coderivative of Mordukhovich), the opening of new
applications (e.g., the calibration of water supply systems), or
the elaboration of new solution algorithms (e.g., smoothing
methods).
The book is divided into three parts. The focus in the first
part is on bilevel programming. The chapters in the second part
contain investigations of mathematical programs with equilibrium
constraints. The third part is on multivalued set-valued
optimization. The chapters were written by outstanding experts in
the areas of bilevel programming, mathematical programs with
equilibrium (or complementarity) constraints (MPEC), and set-valued
optimization problems.
In the field of nondifferentiable nonconvex optimization, one of
the most intensely investigated areas is that of optimization
problems involving multivalued mappings in constraints or as the
objective function. This book focuses on the tremendous development
in the field that has taken place since the publication of the most
recent volumes on the subject. The new topics studied include the
formulation of optimality conditions using different kinds of
generalized derivatives for set-valued mappings (such as, for
example, the coderivative of Mordukhovich), the opening of new
applications (e.g., the calibration of water supply systems), or
the elaboration of new solution algorithms (e.g., smoothing
methods). The book is divided into three parts. The focus in the
first part is on bilevel programming. The chapters in the second
part contain investigations of mathematical programs with
equilibrium constraints. The third part is on multivalued
set-valued optimization. The chapters were written by outstanding
experts in the areas of bilevel programming, mathematical programs
with equilibrium (or complementarity) constraints (MPEC), and
set-valued optimization problems.
Bilevel programming problems are hierarchical optimization problems
where the constraints of one problem (the so-called upper level
problem) are defined in part by a second parametric optimization
problem (the lower level problem). If the lower level problem has a
unique optimal solution for all parameter values, this problem is
equivalent to a one-level optimization problem having an implicitly
defined objective function. Special emphasize in the book is on
problems having non-unique lower level optimal solutions, the
optimistic (or weak) and the pessimistic (or strong) approaches are
discussed. The book starts with the required results in parametric
nonlinear optimization. This is followed by the main theoretical
results including necessary and sufficient optimality conditions
and solution algorithms for bilevel problems. Stationarity
conditions can be applied to the lower level problem to transform
the optimistic bilevel programming problem into a one-level
problem. Properties of the resulting problem are highlighted and
its relation to the bilevel problem is investigated. Stability
properties, numerical complexity, and problems having additional
integrality conditions on the variables are also discussed.
Audience: Applied mathematicians and economists working in
optimization, operations research, and economic modelling. Students
interested in optimization will also find this book useful.
Das Vieweg+Teubner Taschenbuch der Mathematik erfullt aktuell,
umfassend und kompakt alle Erwartungen, die an ein mathematisches
Nachschlagewerk gestellt werden. Es vermittelt ein lebendiges und
modernes Bild der heutigen Mathematik. Als Taschenbuch begleitet es
die Bachelor-Studierenden vom ersten Semester bis zur letzten
Prufung und der Praktiker nutzt es als standiges und
unentbehrliches Nachschlagewerk in seinem Berufsalltag. Das
Taschenbuch bietet alles, was in Bachelor-Studiengangen im Haupt-
und Nebenfach Mathematik benotigt wird. Der Text fur diese Ausgabe
wurde stark uberarbeitet. Zu spezielle Inhalte wurden
herausgenommen und dafur Themen der Wirtschaftsmathematik und
Algorithmik hinzugenommen. Das Vieweg+Teubner Handbuch der
Mathematik (eAusgabe) enthalt daruberhinaus erganzendes und
weiterfuhrendes Material fur das
Masterstudium.
"
Bilevel programming problems are hierarchical optimization problems
where the constraints of one problem (the so-called upper level
problem) are defined in part by a second parametric optimization
problem (the lower level problem). If the lower level problem has a
unique optimal solution for all parameter values, this problem is
equivalent to a one-level optimization problem having an implicitly
defined objective function. Special emphasize in the book is on
problems having non-unique lower level optimal solutions, the
optimistic (or weak) and the pessimistic (or strong) approaches are
discussed. The book starts with the required results in parametric
nonlinear optimization. This is followed by the main theoretical
results including necessary and sufficient optimality conditions
and solution algorithms for bilevel problems. Stationarity
conditions can be applied to the lower level problem to transform
the optimistic bilevel programming problem into a one-level
problem. Properties of the resulting problem are highlighted and
its relation to the bilevel problem is investigated. Stability
properties, numerical complexity, and problems having additional
integrality conditions on the variables are also discussed.
Audience: Applied mathematicians and economists working in
optimization, operations research, and economic modelling. Students
interested in optimization will also find this book useful.
Als mehrbandiges Nachschlagewerk ist das Springer-Handbuch der
Mathematik in erster Linie fur wissenschaftliche Bibliotheken,
akademische Institutionen und Firmen sowie interessierte
Individualkunden in Forschung und Lehregedacht. Es erganzt das
einbandige themenumfassende Springer-Taschenbuch der Mathematik
(ehemaliger Titel Teubner-Taschenbuch der Mathematik), das sich in
seiner begrenzten Stoffauswahl besonders an Studierende richtet.
Teil III des Springer-Handbuchs enthalt neben den Kapiteln 5-9 des
Springer-Taschenbuchs zusatzliches Material zu stochastischen
Prozessen.
Anhand vieler Praxisbeispiele aus den Wirtschaftswissenschaften
bietet dieses Buch einen praktischen Einstieg in die Lineare
Optimierung. Dabei werden dem Leser insbesondere die zugrunde
liegenden Ideen nahe vermittelt. Zu den zahlreichen Aufgaben werden
ausfuhrliche Musterlosungen angeboten. Daruber hinaus helfen Ihnen
die Beispielklausuren bei der gezielten Vorbereitung auf
Klausuren."
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