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This book contains three well-written research tutorials that
inform the graduate reader about the forefront of current research
in multi-agent optimization. These tutorials cover topics that have
not yet found their way in standard books and offer the reader the
unique opportunity to be guided by major researchers in the
respective fields. Multi-agent optimization, lying at the
intersection of classical optimization, game theory, and
variational inequality theory, is at the forefront of modern
optimization and has recently undergone a dramatic development. It
seems timely to provide an overview that describes in detail
ongoing research and important trends. This book concentrates on
Distributed Optimization over Networks; Differential Variational
Inequalities; and Advanced Decomposition Algorithms for Multi-agent
Systems. This book will appeal to both mathematicians and
mathematically oriented engineers and will be the source of
inspiration for PhD students and researchers.
Computational Optimization: A Tribute to Olvi Mangasarian serves as
an excellent reference, providing insight into some of the most
challenging research issues in the field. This collection of papers
covers a wide spectrum of computational optimization topics,
representing a blend of familiar nonlinear programming topics and
such novel paradigms as semidefinite programming and
complementarity-constrained nonlinear programs. Many new results
are presented in these papers which are bound to inspire further
research and generate new avenues for applications. An informal
categorization of the papers includes: * Algorithmic advances for
special classes of constrained optimization problems * Analysis of
linear and nonlinear programs * Algorithmic advances * B-
stationary points of mathematical programs with equilibrium
constraints * Applications of optimization * Some mathematical
topics * Systems of nonlinear equations.
The ?nite-dimensional nonlinear complementarity problem (NCP) is a
s- tem of ?nitely many nonlinear inequalities in ?nitely many
nonnegative variables along with a special equation that expresses
the complementary relationship between the variables and
corresponding inequalities. This complementarity condition is the
key feature distinguishing the NCP from a general inequality
system, lies at the heart of all constrained optimi- tion problems
in ?nite dimensions, provides a powerful framework for the modeling
of equilibria of many kinds, and exhibits a natural link between
smooth and nonsmooth mathematics. The ?nite-dimensional variational
inequality (VI), which is a generalization of the NCP, provides a
broad unifying setting for the study of optimization and
equilibrium problems and serves as the main computational framework
for the practical solution of a host of continuum problems in the
mathematical sciences. The systematic study of the
?nite-dimensional NCP and VI began in the mid-1960s; in a span of
four decades, the subject has developed into a very fruitful
discipline in the ?eld of mathematical programming. The -
velopments include a rich mathematical theory, a host of e?ective
solution algorithms, a multitude of interesting connections to
numerous disciplines, and a wide range of important applications in
engineering and economics. As a result of their broad associations,
the literature of the VI/CP has bene?ted from contributions made by
mathematicians (pure, applied, and computational), computer
scientists, engineers of many kinds (civil, ch- ical, electrical,
mechanical, and systems), and economists of diverse exp- tise
(agricultural, computational, energy, ?nancial, and spatial).
The ?nite-dimensional nonlinear complementarity problem (NCP) is a
s- tem of ?nitely many nonlinear inequalities in ?nitely many
nonnegative variables along with a special equation that expresses
the complementary relationship between the variables and
corresponding inequalities. This complementarity condition is the
key feature distinguishing the NCP from a general inequality
system, lies at the heart of all constrained optimi- tion problems
in ?nite dimensions, provides a powerful framework for the modeling
of equilibria of many kinds, and exhibits a natural link between
smooth and nonsmooth mathematics. The ?nite-dimensional variational
inequality (VI), which is a generalization of the NCP, provides a
broad unifying setting for the study of optimization and
equilibrium problems and serves as the main computational framework
for the practical solution of a host of continuum problems in the
mathematical sciences. The systematic study of the
?nite-dimensional NCP and VI began in the mid-1960s; in a span of
four decades, the subject has developed into a very fruitful
discipline in the ?eld of mathematical programming. The -
velopments include a rich mathematical theory, a host of e?ective
solution algorithms, a multitude of interesting connections to
numerous disciplines, and a wide range of important applications in
engineering and economics. As a result of their broad associations,
the literature of the VI/CP has bene?ted from contributions made by
mathematicians (pure, applied, and computational), computer
scientists, engineers of many kinds (civil, ch- ical, electrical,
mechanical, and systems), and economists of diverse exp- tise
(agricultural, computational, energy, ?nancial, and spatial).
This volume presents state-of-the-art complementarity applications,
algorithms, extensions and theory in the form of eighteen papers.
These at the International Conference on Com invited papers were
presented plementarity 99 (ICCP99) held in Madison, Wisconsin
during June 9-12, 1999 with support from the National Science
Foundation under Grant DMS-9970102. Complementarity is becoming
more widely used in a variety of appli cation areas. In this
volume, there are papers studying the impact of complementarity in
such diverse fields as deregulation of electricity mar kets,
engineering mechanics, optimal control and asset pricing. Further
more, application of complementarity and optimization ideas to
related problems in the burgeoning fields of machine learning and
data mining are also covered in a series of three articles. In
order to effectively process the complementarity problems that
arise in such applications, various algorithmic, theoretical and
computational extensions are covered in this volume. Nonsmooth
analysis has an im portant role to play in this area as can be seen
from articles using these tools to develop Newton and path
following methods for constrained nonlinear systems and
complementarity problems. Convergence issues are covered in the
context of active set methods, global algorithms for pseudomonotone
variational inequalities, successive convex relaxation and proximal
point algorithms. Theoretical contributions to the connectedness of
solution sets and constraint qualifications in the growing area of
mathematical programs with equilibrium constraints are also
presented. A relaxation approach is given for solving such
problems. Finally, computational issues related to preprocessing
mixed complementarity problems are addressed."
This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.
This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.
This volume presents state-of-the-art complementarity applications,
algorithms, extensions and theory in the form of eighteen papers.
These at the International Conference on Com invited papers were
presented plementarity 99 (ICCP99) held in Madison, Wisconsin
during June 9-12, 1999 with support from the National Science
Foundation under Grant DMS-9970102. Complementarity is becoming
more widely used in a variety of appli cation areas. In this
volume, there are papers studying the impact of complementarity in
such diverse fields as deregulation of electricity mar kets,
engineering mechanics, optimal control and asset pricing. Further
more, application of complementarity and optimization ideas to
related problems in the burgeoning fields of machine learning and
data mining are also covered in a series of three articles. In
order to effectively process the complementarity problems that
arise in such applications, various algorithmic, theoretical and
computational extensions are covered in this volume. Nonsmooth
analysis has an im portant role to play in this area as can be seen
from articles using these tools to develop Newton and path
following methods for constrained nonlinear systems and
complementarity problems. Convergence issues are covered in the
context of active set methods, global algorithms for pseudomonotone
variational inequalities, successive convex relaxation and proximal
point algorithms. Theoretical contributions to the connectedness of
solution sets and constraint qualifications in the growing area of
mathematical programs with equilibrium constraints are also
presented. A relaxation approach is given for solving such
problems. Finally, computational issues related to preprocessing
mixed complementarity problems are addressed."
Computational Optimization: A Tribute to Olvi Mangasarian serves as
an excellent reference, providing insight into some of the most
challenging research issues in the field. This collection of papers
covers a wide spectrum of computational optimization topics,
representing a blend of familiar nonlinear programming topics and
such novel paradigms as semidefinite programming and
complementarity-constrained nonlinear programs. Many new results
are presented in these papers which are bound to inspire further
research and generate new avenues for applications. An informal
categorization of the papers includes: Algorithmic advances for
special classes of constrained optimization problems Analysis of
linear and nonlinear programs Algorithmic advances B- stationary
points of mathematical programs with equilibrium constraints
Applications of optimization Some mathematical topics Systems of
nonlinear equations.
Starting with the fundamentals of classical smooth optimization and
building on established convex programming techniques, this
research monograph presents a foundation and methodology for modern
nonconvex nondifferentiable optimization. It provides readers with
theory, methods, and applications of nonconvex and
nondifferentiable optimization in statistical estimation,
operations research, machine learning, and decision making. A
comprehensive and rigorous treatment of this emergent mathematical
topic is urgently needed in today's complex world of big data and
machine learning. This book takes a thorough approach to the
subject and includes examples and exercises to enrich the main
themes, making it suitable for classroom instruction. Modern
Nonconvex Nondifferentiable Optimization is intended for applied
and computational mathematicians, optimizers, operations
researchers, statisticians, computer scientists, engineers,
economists, and machine learners. It could be used in advanced
courses on optimization/operations research and nonconvex and
nonsmooth optimization.
Awarded the Frederick W. Lanchester Prize in 1994 for its valuable
contributions to operations research and the management sciences,
this mathematically rigorous book remains the standard reference on
the linear complementarity problem. Its comprehensive treatment of
the computation of equilibria arising from engineering, economics,
and finance, plus chapter-ending exercises and 'Notes and
References' sections make it equally useful for a graduate-level
course or for self-study. For this new edition the authors have
corrected typographical errors, revised difficult or faulty
passages, and updated the bibliography.
This book provides a solid foundation and an extensive study for an
important class of constrained optimization problems known as
Mathematical Programs with Equilibrium Constraints (MPEC), which
are extensions of bilevel optimization problems. The book begins
with the description of many source problems arising from
engineering and economics that are amenable to treatment by the
MPEC methodology. Error bounds and parametric analysis are the main
tools to establish a theory of exact penalisation, a set of MPEC
constraint qualifications and the first-order and second-order
optimality conditions. The book also describes several iterative
algorithms such as a penalty-based interior point algorithm, an
implicit programming algorithm and a piecewise sequential quadratic
programming algorithm for MPECs. Results in the book are expected
to have significant impacts in such disciplines as engineering
design, economics and game equilibria, and transportation planning,
within all of which MPEC has a central role to play in the
modelling of many practical problems.
This book provides a solid foundation and an extensive study for Mathematical Programs with Equilibrium Constraints (MPEC). It begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalization, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modeling of many practical problems.
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