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The Moment-SOS hierarchy is a powerful methodology that is used to
solve the Generalized Moment Problem (GMP) where the list of
applications in various areas of Science and Engineering is almost
endless. Initially designed for solving polynomial optimization
problems (the simplest example of the GMP), it applies to solving
any instance of the GMP whose description only involves
semi-algebraic functions and sets. It consists of solving a
sequence (a hierarchy) of convex relaxations of the initial
problem, and each convex relaxation is a semidefinite program whose
size increases in the hierarchy.The goal of this book is to
describe in a unified and detailed manner how this methodology
applies to solving various problems in different areas ranging from
Optimization, Probability, Statistics, Signal Processing,
Computational Geometry, Control, Optimal Control and Analysis of a
certain class of nonlinear PDEs. For each application, this
unconventional methodology differs from traditional approaches and
provides an unusual viewpoint. Each chapter is devoted to a
particular application, where the methodology is thoroughly
described and illustrated on some appropriate examples.The
exposition is kept at an appropriate level of detail to aid the
different levels of readers not necessarily familiar with these
tools, to better know and understand this methodology.
Many important problems in global optimization, algebra,
probability and statistics, applied mathematics, control theory,
financial mathematics, inverse problems, etc. can be modeled as a
particular instance of the Generalized Moment Problem (GMP). This
book introduces, in a unified manual, a new general methodology to
solve the GMP when its data are polynomials and basic
semi-algebraic sets. This methodology combines semidefinite
programming with recent results from real algebraic geometry to
provide a hierarchy of semidefinite relaxations converging to the
desired optimal value. Applied on appropriate cones, standard
duality in convex optimization nicely expresses the duality between
moments and positive polynomials. In the second part of this
invaluable volume, the methodology is particularized and described
in detail for various applications, including global optimization,
probability, optimal context, mathematical finance, multivariate
integration, etc., and examples are provided for each particular
application.
Many important applications in global optimization, algebra,
probability and statistics, applied mathematics, control theory,
financial mathematics, inverse problems, etc. can be modeled as a
particular instance of the Generalized Moment Problem (GMP).This
book introduces a new general methodology to solve the GMP when its
data are polynomials and basic semi-algebraic sets. This
methodology combines semidefinite programming with recent results
from real algebraic geometry to provide a hierarchy of semidefinite
relaxations converging to the desired optimal value. Applied on
appropriate cones, standard duality in convex optimization nicely
expresses the duality between moments and positive polynomials.In
the second part, the methodology is particularized and described in
detail for various applications, including global optimization,
probability, optimal control, mathematical finance, multivariate
integration, etc., and examples are provided for each particular
application.
This book aims at gathering roboticists, control theorists,
neuroscientists, and mathematicians, in order to promote a
multidisciplinary research on movement analysis. It follows the
workshop " Geometric and Numerical Foundations of Movements " held
at LAAS-CNRS in Toulouse in November 2015[1]. Its objective is to
lay the foundations for a mutual understanding that is essential
for synergetic development in motion research. In particular, the
book promotes applications to robotics --and control in general--
of new optimization techniques based on recent results from real
algebraic geometry.
This book aims at gathering roboticists, control theorists,
neuroscientists, and mathematicians, in order to promote a
multidisciplinary research on movement analysis. It follows the
workshop " Geometric and Numerical Foundations of Movements " held
at LAAS-CNRS in Toulouse in November 2015[1]. Its objective is to
lay the foundations for a mutual understanding that is essential
for synergetic development in motion research. In particular, the
book promotes applications to robotics --and control in general--
of new optimization techniques based on recent results from real
algebraic geometry.
Integer programming (IP) is a fascinating topic. Indeed, while
linear programming (LP), its c- tinuous analogue, is well
understood and extremely ef?cient LP software packages exist,
solving an integer program can remain a formidable challenge, even
for some small size problems. For instance, the following small
(5-variable) IP problem (called the unbounded knapsack problem)
min{213x?1928x?11111x?2345x +9123x} 1 2 3 4 5 s.t. 12223x +12224x
+36674x +61119x +85569x = 89643482, 1 2 3 4 5 x ,x ,x ,x ,x?N, 1 2
3 4 5 taken from a list of dif?cult knapsack problems in Aardal and
Lenstra [2], is not solved even by hours of computing, using for
instance the last version of the ef?cient software package CPLEX.
However,thisisnotabookonintegerprogramming,asverygoodonesonthistopicalreadyexist.
For standard references on the theory and practice of integer
programming, the interested reader is referred to, e.g., Nemhauser
and Wolsey [113], Schrijver [121], Wolsey [136], and the more
recent Bertsimas and Weismantel [21]. On the other hand, this book
could provide a complement to the above books as it develops a
rather unusual viewpoint.
Production Management is a large field concerned with all the
aspects related to production, from the very bottom decisions at
the machine level, to the top-level strategic decisicns. In this
book, we are concerned with production planning and scheduling
aspects. Traditional production planning methodologies are based on
a now widely ac cepted hierarchical decom?osition into several
planning decision levels. The higher in the hierarchy, the more
aggregate are the models and the more important are the decisions.
In this book, we only consider the last two decision levels in the
hierarchy, namely, the mid-term (or tacticaQ planning level and the
short-term (or operationaQ scheduling level. In the literature and
in practice, the decisions are taken in sequence and in a top-down
approach from the highest level in the hierarchy to the bottom
level. The decisions taken at some level in the hierarchy are
constrained by those already taken at upper levels and in turn,
must translate into feasible objectives for the next lower levels
in the hierarchy. It is a common sense remark to say that the whole
hierarchical decision process is coherent if the interactions
between different levels in the hierarchy are taken into account so
that a decision taken at some level in the hierarchy translates
into a feasible objective for the next decision level in the
hierarchy. However, and surpris ingly enough, this crucial
consistency issue is rarely investigated and few results are
available in the literature."
The Christoffel-Darboux kernel, a central object in approximation
theory, is shown to have many potential uses in modern data
analysis, including applications in machine learning. This is the
first book to offer a rapid introduction to the subject,
illustrating the surprising effectiveness of a simple tool.
Bridging the gap between classical mathematics and current evolving
research, the authors present the topic in detail and follow a
heuristic, example-based approach, assuming only a basic background
in functional analysis, probability and some elementary notions of
algebraic geometry. They cover new results in both pure and applied
mathematics and introduce techniques that have a wide range of
potential impacts on modern quantitative and qualitative science.
Comprehensive notes provide historical background, discuss advanced
concepts and give detailed bibliographical references. Researchers
and graduate students in mathematics, statistics, engineering or
economics will find new perspectives on traditional themes, along
with challenging open problems.
This is the first comprehensive introduction to the powerful moment
approach for solving global optimization problems (and some related
problems) described by polynomials (and even semi-algebraic
functions). In particular, the author explains how to use
relatively recent results from real algebraic geometry to provide a
systematic numerical scheme for computing the optimal value and
global minimizers. Indeed, among other things, powerful positivity
certificates from real algebraic geometry allow one to define an
appropriate hierarchy of semidefinite (SOS) relaxations or LP
relaxations whose optimal values converge to the global minimum.
Several extensions to related optimization problems are also
described. Graduate students, engineers and researchers entering
the field can use this book to understand, experiment with and
master this new approach through the simple worked examples
provided.
This is the first comprehensive introduction to the powerful moment
approach for solving global optimization problems (and some related
problems) described by polynomials (and even semi-algebraic
functions). In particular, the author explains how to use
relatively recent results from real algebraic geometry to provide a
systematic numerical scheme for computing the optimal value and
global minimizers. Indeed, among other things, powerful positivity
certificates from real algebraic geometry allow one to define an
appropriate hierarchy of semidefinite (SOS) relaxations or LP
relaxations whose optimal values converge to the global minimum.
Several extensions to related optimization problems are also
described. Graduate students, engineers and researchers entering
the field can use this book to understand, experiment with and
master this new approach through the simple worked examples
provided.
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