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Algorithms for Solving Common Fixed Point Problems
$109.00
Springer
May-30-2018

This book details approximate solutions to common fixed point problems and convex feasibility problems in the presence of perturbations. Convex feasibility problems search for a common point of a finite collection of subsets in a Hilbert space; common fixed point problems pursue a common fixed point of a finite collection of self-mappings in a Hilbert space. A variety of algorithms are considered in this book for solving both types of problems,  the study of which has fueled a rapidly growing area of research. This monograph is timely and highlights the numerous applications to engineering, computed tomography, and radiation therapy planning.

Totaling eight chapters, this book begins with an introduction to foundational material and moves on to examine iterative methods in metric spaces. The dynamic string-averaging methods for common fixed point problems in normed space are analyzed in Chapter 3. Dynamic string methods, for common fixed point problems in a metric space are introduced and discussed in Chapter 4. Chapter 5 is devoted to the convergence of an abstract version of the algorithm which has been called  component-averaged row projections (CARP). Chapter 6 studies a proximal algorithm for finding a common zero of a family of maximal monotone operators. Chapter 7 extends the results of Chapter 6 for a dynamic string-averaging version of the proximal algorithm. In Chapters 8 subgradient projections algorithms for convex feasibility problems are examined for infinite dimensional Hilbert spaces. 

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Discrete-Time Optimal Control and Games on Large Intervals
$130.13
Springer
April-10-2017

Devoted to the structure of approximate solutions of discrete-time optimal control problems and approximate solutions of dynamic discrete-time two-player zero-sum games, this book presents results on properties of approximate solutions in an interval that is independent lengthwise, for all sufficiently large intervals. Results concerning the so-called turnpike property of optimal control problems and zero-sum games in the regions close to the endpoints of the time intervals are the main focus of this book. The description of the structure of approximate solutions on sufficiently large intervals and its stability will interest graduate students and mathematicians in optimal control and game theory, engineering, and economics.

This book begins with a brief overview and moves on to analyze the structure of approximate solutions of autonomous nonconcave discrete-time optimal control Lagrange problems.Next the structures of approximate solutions of autonomous discrete-time optimal control problems that are discrete-time analogs of Bolza problems in calculus of variations are studied. The structures of approximate solutions of two-player zero-sum games are analyzed through standard convexity-concavity assumptions. Finally, turnpike properties for approximate solutions in a class of nonautonomic dynamic discrete-time games with convexity-concavity assumptions are examined.   

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Approximate Solutions of Common Fixed-Point Problems
$130.13
Springer
July-08-2016

This book presents results on the convergence behavior of algorithms which are known as vital tools for solving convex feasibility problems and common fixed point problems. The main goal for us in dealing with a known computational error is to find what approximate solution can be obtained and how many iterates one needs to find it. According to know results, these algorithms should converge to a solution. In this exposition, these algorithms are studied, taking into account computational errors which remain consistent in practice. In this case the convergence to a solution does not take place. We show that our algorithms generate a good approximate solution if computational errors are bounded from above by a small positive constant.

Beginning  with an introduction, this monograph moves on to study:

� dynamic string-averaging methods for common fixed point problems in a Hilbert space

� dynamic string methods for common fixed point problems in a metric space<

� dynamic string-averaging version of the proximal algorithm

� common fixed point problems in metric spaces

� common fixed point problems in the spaces with distances of the Bregman type

� a proximal algorithm for finding a common zero of a family of maximal monotone operators

� subgradient projections algorithms for convex feasibility problems in Hilbert spaces 

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Numerical Optimization with Computational Errors
$114.73
Springer
May-03-2016

This book studies the approximate solutions of optimization problems in the presence of computational errors. A number of results are presented on the convergence behavior of algorithms in a Hilbert space; these algorithms are examined taking into account computational errors. The author illustrates that algorithms generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Known computational errors  are examined with the aim of determining an approximate solution. Researchers and students interested in the optimization theory and its applications will find this book instructive and informative.

 

This monograph contains 16 chapters; including a chapters devoted to the subgradient projection algorithm, the mirror descent algorithm, gradient projection algorithm, the Weiszfelds method, constrained convex minimization problems, the convergence of a proximal point method in a Hilbert space, the continuous subgradient method, penalty methods and Newton?s method.

  

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Turnpike Theory of Continuous-Time Linear Optimal Control Problems
$99.33
Springer
July-13-2015

Individual turnpike results are of great interest due to their numerous applications in engineering and in economic theory; in this book the study is focused on new results of turnpike phenomenon in linear optimal control problems. The book is intended for engineers as well as for mathematicians interested in the calculus of variations, optimal control and in applied functional analysis.

Two large classes of problems are studied in more depth. The first class studied in Chapter 2 consists of linear control problems with periodic nonsmooth convex integrands. Chapters 3-5 consist of linear control problems with autonomous convex smooth integrands.� Chapter 6 discusses a turnpike property for dynamic zero-sum games with linear constraints. Chapter 7 examines genericity results. In Chapter 8, the description of structure of variational problems with extended-valued integrands is obtained. Chapter 9 ends the exposition with a study of turnpike phenomenon for dynamic games with extended value integrands.

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Turnpike Phenomenon and Infinite Horizon Optimal Control
$114.73
Springer
September-17-2014

This book is devoted to the study of the turnpike phenomenon and describes the existence of solutions for a large variety of infinite horizon optimal control classes of problems. Chapter 1 provides introductory material on turnpike properties. Chapter 2 studies the turnpike phenomenon for discrete-time optimal control problems. The turnpike properties of autonomous problems with extended-value integrands are studied in Chapter 3. Chapter 4 focuses on large classes of infinite horizon optimal control problems without convexity (concavity) assumptions. In Chapter 5, the turnpike results for a class of dynamic discrete-time two-player zero-sum game are proven.

This thorough exposition will be very useful�for mathematicians working in the fields of optimal control, the calculus of variations, applied functional analysis and infinite horizon optimization. It may also be used as a primary text in a graduate course in optimal control or as supplementary text for a variety of courses in other disciplines. Researchers in other fields such as economics and game theory, where turnpike properties are well known, will also find this Work valuable.

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Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems
$57.74
Springer
September-01-2014
The structure of approximate solutions of autonomous discrete-time optimal control problems and individual turnpike results for optimal control problems without convexity (concavity) assumptions are examined in this book. In particular, the book focuses on the properties of approximate solutions which are independent of the length of the interval, for all sufficiently large intervals; these results apply to the so-called turnpike property of the optimal control problems. By encompassing the so-called turnpike property the approximate solutions of the problems�are determined primarily by the objective function and are fundamentally independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. This book�also explores the turnpike phenomenon for two large classes of autonomous optimal control problems. It is illustrated that the turnpike phenomenon is stable for an optimal control problem if the corresponding infinite horizon optimal control problem possesses an asymptotic turnpike property. If an optimal control problem belonging to the first class possesses the turnpike property, then the turnpike is a singleton (unit set). The stability of the turnpike property under small perturbations of an objective function and of a constraint map is established. For the second class of problems where the turnpike phenomenon is not necessarily a singleton the stability of the turnpike property under small perturbations of an objective function is established. Containing solutions of difficult problems in optimal control�and presenting new approaches, techniques and methods this book is of interest for�mathematicians�working in optimal control and the calculus of variations.�It also can be useful in preparation courses for graduate students.
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Structure of Approximate Solutions of Optimal Control Problems
$57.74
Springer
August-16-2013
This title�examines the structure of approximate solutions of optimal control problems considered on subintervals of a real line. Specifically at the properties of approximate solutions which are independent of the length of the interval. The results illustrated in this book look into the so-called turnpike property of optimal control problems. �The author generalizes the�results�of the turnpike property by considering �a class of optimal control problems which is identified with the corresponding complete metric space of objective functions.�This establishes the turnpike property for any element in a set that is in�a countable intersection�which is open everywhere dense sets in the space of integrands; meaning that the turnpike property holds for most optimal control problems. Mathematicians working in optimal control and the calculus of variations and graduate students will find this book��useful and valuable due to its� presentation of solutions to a number of difficult problems in optimal control��and presentation of new approaches, techniques and methods.
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Nonconvex Optimal Control and Variational Problems
$153.23
Springer
June-11-2013

Nonconvex Optimal Control and Variational Problems is an important contribution to the existing literature in the field and is devoted to the presentation of progress made in the last 15 years of research in the area of optimal control and the calculus of variations. This volume contains a number of results concerning well-posedness of optimal control and variational problems, nonoccurrence of the Lavrentiev phenomenon for optimal control and variational problems, and turnpike properties of approximate solutions of variational problems.

Chapter 1 contains an introduction as well as examples of select topics. Chapters 2-5 consider the well-posedness condition using fine tools of general topology and porosity. Chapters 6-8 are devoted to the nonoccurrence of the Lavrentiev phenomenon and contain original results. Chapter 9 focuses on infinite-dimensional linear control problems, and Chapter 10 deals with ?good? functions and explores new understandings on the questions of optimality and variational problems. Finally, Chapters 11-12 are centered around the turnpike property, a particular area of expertise for the author.

This volume is intended for mathematicians, engineers, and scientists interested in the calculus of variations, optimal control, optimization, and applied functional analysis, as well as both undergraduate and graduate students specializing in those areas. The text devoted to Turnpike properties may be of particular interest to the economics community.

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Structure of Solutions of Variational Problems
$57.74
Springer
February-09-2013
?Structure of Solutions of Variational Problems is devoted to recent progress made in the studies of the structure of approximate solutions of variational problems considered on subintervals of a real line. �Results on properties of approximate solutions which are independent of the length of the interval, for all sufficiently large intervals are presented in a clear manner. Solutions, new approaches, techniques and methods to a number of difficult problems in the calculus of variations� are illustrated throughout this book. This book also contains significant results and information about the turnpike property of the variational problems.�This well-known property is a general phenomenon which holds for large classes of variational problems. The author examines the following in relation to the turnpike property �in individual� (non-generic) turnpike results, sufficient and necessary conditions for the turnpike phenomenon as well as in the non-intersection property for extremals of variational problems. This book appeals to mathematicians �working in optimal control and the calculus as� well as with graduate students.???
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Optimization on Metric and Normed Spaces
$191.73
Springer
August-17-2010
Optimization on Metric and Normed Spaces is devoted to the recent progress in optimization on Banach spaces and complete metric spaces. Optimization problems are usually considered on metric spaces satisfying certain compactness assumptions which guarantee the existence of solutions and convergence of algorithms. This book considers spaces that do not satisfy such compactness assumptions. In order to overcome these difficulties, the book uses the Baire category approach and considers approximate solutions. Therefore, it presents a number of new results concerning penalty methods in constrained optimization, existence of solutions in parametric optimization, well-posedness of vector minimization problems, and many other results obtained in the last ten years. The book is intended for mathematicians interested in optimization and applied functional analysis.
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Turnpike Properties in the Calculus of Variations and Optimal Control
$168.63
Springer
August-25-2005