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This book contains extended, in-depth presentations of the plenary
talks from the 16th French-German-Polish Conference on
Optimization, held in Krakow, Poland in 2013. Each chapter in this
book exhibits a comprehensive look at new theoretical and/or
application-oriented results in mathematical modeling,
optimization, and optimal control. Students and researchers
involved in image processing, partial differential inclusions,
shape optimization, or optimal control theory and its applications
to medical and rehabilitation technology, will find this book
valuable. The first chapter by Martin Burger provides an overview
of recent developments related to Bregman distances, which is an
important tool in inverse problems and image processing. The
chapter by Piotr Kalita studies the operator version of a first
order in time partial differential inclusion and its time
discretization. In the chapter by Gunter Leugering, Jan Sokolowski
and Antoni Zochowski, nonsmooth shape optimization problems for
variational inequalities are considered. The next chapter, by Katja
Mombaur is devoted to applications of optimal control and inverse
optimal control in the field of medical and rehabilitation
technology, in particular in human movement analysis, therapy and
improvement by means of medical devices. The final chapter, by
Nikolai Osmolovskii and Helmut Maurer provides a survey on no-gap
second order optimality conditions in the calculus of variations
and optimal control, and a discussion of their further development.
This textbook presents a collection of interesting and sometimes
original exercises for motivated students in mathematics. Written
in the same spirit as Volume 1, this second volume of Mathematical
Tapas includes carefully selected problems at the intersection
between undergraduate and graduate level. Hints, answers and
(sometimes) comments are presented alongside the 222 "tapas" as
well as 8 conjectures or open problems. Topics covered include
metric, normed, Banach, inner-product and Hilbert spaces;
differential calculus; integration; matrices; convexity; and
optimization or variational problems. Suitable for advanced
undergraduate and graduate students in mathematics, this book aims
to sharpen the reader's mathematical problem solving abilities.
This book contains a collection of exercises (called "tapas") at
undergraduate level, mainly from the fields of real analysis,
calculus, matrices, convexity, and optimization. Most of the
problems presented here are non-standard and some require broad
knowledge of different mathematical subjects in order to be solved.
The author provides some hints and (partial) answers and also puts
these carefully chosen exercises into context, presents information
on their origins, and comments on possible extensions. With stars
marking the levels of difficulty, these tapas show or prove
something interesting, challenge the reader to solve and learn, and
may have surprising results. This first volume of Mathematical
Tapas will appeal to mathematicians, motivated undergraduate
students from science-based areas, and those generally interested
in mathematics.
This book contains extended, in-depth presentations of the plenary
talks from the 16th French-German-Polish Conference on
Optimization, held in Krakow, Poland in 2013. Each chapter in this
book exhibits a comprehensive look at new theoretical and/or
application-oriented results in mathematical modeling,
optimization, and optimal control. Students and researchers
involved in image processing, partial differential inclusions,
shape optimization, or optimal control theory and its applications
to medical and rehabilitation technology, will find this book
valuable. The first chapter by Martin Burger provides an overview
of recent developments related to Bregman distances, which is an
important tool in inverse problems and image processing. The
chapter by Piotr Kalita studies the operator version of a first
order in time partial differential inclusion and its time
discretization. In the chapter by Gunter Leugering, Jan Sokolowski
and Antoni Zochowski, nonsmooth shape optimization problems for
variational inequalities are considered. The next chapter, by Katja
Mombaur is devoted to applications of optimal control and inverse
optimal control in the field of medical and rehabilitation
technology, in particular in human movement analysis, therapy and
improvement by means of medical devices. The final chapter, by
Nikolai Osmolovskii and Helmut Maurer provides a survey on no-gap
second order optimality conditions in the calculus of variations
and optimal control, and a discussion of their further development.
Convex Analysis may be considered as a refinement of standard
calculus, with equalities and approximations replaced by
inequalities. As such, it can easily be integrated into a graduate
study curriculum. Minimization algorithms, more specifically those
adapted to non-differentiable functions, provide an immediate
application of convex analysis to various fields related to
optimization and operations research. These two topics making up
the title of the book, reflect the two origins of the authors, who
belong respectively to the academic world and to that of
applications. Part I can be used as an introductory textbook (as a
basis for courses, or for self-study); Part II continues this at a
higher technical level and is addressed more to specialists,
collecting results that so far have not appeared in books.
From the reviews: "The account is quite detailed and is written in
a manner that will appeal to analysts and numerical practitioners
alike...they contain everything from rigorous proofs to tables of
numerical calculations.... one of the strong features of these
books...that they are designed not for the expert, but for those
who whish to learn the subject matter starting from little or no
background...there are numerous examples, and counter-examples, to
back up the theory...To my knowledge, no other authors have given
such a clear geometric account of convex analysis." "This
innovative text is well written, copiously illustrated, and
accessible to a wide audience"
Convex Analysis may be considered as a refinement of standard
calculus, with equalities and approximations replaced by
inequalities. As such, it can easily be integrated into a graduate
study curriculum. Minimization algorithms, more specifically those
adapted to non-differentiable functions, provide an immediate
application of convex analysis to various fields related to
optimization and operations research. These two topics making up
the title of the book, reflect the two origins of the authors, who
belong respectively to the academic world and to that of
applications. Part I can be used as an introductory textbook (as a
basis for courses, or for self-study); Part II continues this at a
higher technical level and is addressed more to specialists,
collecting results that so far have not appeared in books.
From the reviews: "The account is quite detailed and is written in
a manner that will appeal to analysts and numerical practitioners
alike...they contain everything from rigorous proofs to tables of
numerical calculations.... one of the strong features of these
books...that they are designed not for the expert, but for those
who whish to learn the subject matter starting from little or no
background...there are numerous examples, and counter-examples, to
back up the theory...To my knowledge, no other authors have given
such a clear geometric account of convex analysis." "This
innovative text is well written, copiously illustrated, and
accessible to a wide audience"
This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and a study of convex minimization problems. The "backbone" of both volumes was extracted, some material deleted that was deemed too advanced for an introduction, or too closely related to numerical algorithms. Some exercises were included and finally the index has been considerably enriched. The main motivation of the authors was to "light the entrance" of the monument Convex Analysis. This book is not a reference book to be kept on the shelf by experts who already know the building and can find their way through it; it is far more a book for the purpose of learning and teaching.
L'etude mathematique des problemes d'optimisation, ou de ceux dits
variationnels de maniere generale (c'est-a-dire, " toute situation
ou il y a quelque chose a minimiser sous des contraintes "),
requiert en prealable qu'on en maitrise les bases, les outils
fondamentaux et quelques principes. Le present ouvrage est un cours
repondant en partie a cette demande, il est principalement destine
a des etudiants de Master en formation, et restreint a l'essentiel.
Sont abordes successivement : La semicontinuite inferieure, les
topologies faibles, les resultats fondamentaux d'existence en
optimisation ; Les conditions d'optimalite approchee ; Des
developpements sur la projection sur un convexe ferme, notamment
sur un cone convexe ferme ; L'analyse convexe dans son role
operatoire ; Quelques schemas de dualisation dans des problemes
d'optimisation non convexe structures ; Une introduction aux
sous-differentiels generalises de fonctions non differentiables.
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