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Convex Optimization with Computational Errors
The book is devoted to the study of approximate solutions of optimization problems in the presence of computational errors. It contains a number of results on the convergence behavior of algorithms in a Hilbert space, which are known as important tools for solving optimization problems. The research presented in the book is the continuation and the further development of the author's (c) 2016 book Numerical Optimization with Computational Errors, Springer 2016. Both books study the algorithms taking into account computational errors which are always present in practice. The main goal is, for a known computational error, to find out what an approximate solution can be obtained and how many it...
Turnpike Phenomenon and Symmetric Optimization Problems
Written by a leading expert in turnpike phenomenon, this book is devoted to the study of symmetric optimization, variational and optimal control problems in infinite dimensional spaces and turnpike properties of their approximate solutions. The book presents a systematic and comprehensive study of general classes of problems in optimization, calculus of variations, and optimal control with symmetric structures from the viewpoint of the turnpike phenomenon. The author establishes generic existence and well-posedness results for optimization problems and individual (not generic) turnpike results for variational and optimal control problems. Rich in impressive theoretical results, the author presents applications to crystallography and discrete dispersive dynamical systems which have prototypes in economic growth theory. This book will be useful for researchers interested in optimal control, calculus of variations turnpike theory and their applications, such as mathematicians, mathematical economists, and researchers in crystallography, to name just a few.
Solutions of Fixed Point Problems with Computational Errors
The book is devoted to the study of approximate solutions of fixed point problems in the presence of computational errors. It begins with a study of approximate solutions of star-shaped feasibility problems in the presence of perturbations. The goal is to show the convergence of algorithms, which are known as important tools for solving convex feasibility problems and common fixed point problems.The text also presents studies of algorithms based on unions of nonexpansive maps, inconsistent convex feasibility problems, and split common fixed point problems. A number of algorithms are considered for solving convex feasibility problems and common fixed point problems. The book will be of interest for researchers and engineers working in optimization, numerical analysis, and fixed point theory. It also can be useful in preparation courses for graduate students. The main feature of the book which appeals specifically to this audience is the study of the influence of computational errors for several important algorithms used for nonconvex feasibility problems.
Optimization on Solution Sets of Common Fixed Point Problems
This book is devoted to a detailed study of the subgradient projection method and its variants for convex optimization problems over the solution sets of common fixed point problems and convex feasibility problems. These optimization problems are investigated to determine good solutions obtained by different versions of the subgradient projection algorithm in the presence of sufficiently small computational errors. The use of selected algorithms is highlighted including the Cimmino type subgradient, the iterative subgradient, and the dynamic string-averaging subgradient. All results presented are new. Optimization problems where the underlying constraints are the solution sets of other problems, frequently occur in applied mathematics. The reader should not miss the section in Chapter 1 which considers some examples arising in the real world applications. The problems discussed have an important impact in optimization theory as well. The book will be useful for researches interested in the optimization theory and its applications.
The Projected Subgradient Algorithm in Convex Optimization
This focused monograph presents a study of subgradient algorithms for constrained minimization problems in a Hilbert space. The book is of interest for experts in applications of optimization to engineering and economics. The goal is to obtain a good approximate solution of the problem in the presence of computational errors. The discussion takes into consideration the fact that for every algorithm its iteration consists of several steps and that computational errors for different steps are different, in general. The book is especially useful for the reader because it contains solutions to a number of difficult and interesting problems in the numerical optimization. The subgradient projectio...
Nonconvex Optimal Control and Variational Problems
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 topol...
Optimization in Banach Spaces
The book is devoted to the study of constrained minimization problems on closed and convex sets in Banach spaces with a Frechet differentiable objective function. Such problems are well studied in a finite-dimensional space and in an infinite-dimensional Hilbert space. When the space is Hilbert there are many algorithms for solving optimization problems including the gradient projection algorithm which is one of the most important tools in the optimization theory, nonlinear analysis and their applications. An optimization problem is described by an objective function and a set of feasible points. For the gradient projection algorithm each iteration consists of two steps. The first step is a ...
Numerical Optimization with Computational Errors
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.
Optimization on Metric and Normed Spaces
"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.
Turnpike Phenomenon in Metric Spaces
This book is devoted to the study of the turnpike phenomenon arising in optimal control theory. Special focus is placed on Turnpike results, in sufficient and necessary conditions for the turnpike phenomenon and in its stability under small perturbations of objective functions. The most important feature of this book is that it develops a large, general class of optimal control problems in metric space. Additional value is in the provision of solutions to a number of difficult and interesting problems in optimal control theory in metric spaces. Mathematicians working in optimal control, optimization, and experts in applications of optimal control to economics and engineering, will find this ...
