Optimization in Function Spaces

Optimization in Function Spaces
Author: Peter Kosmol
Publisher: Walter de Gruyter
Total Pages: 405
Release: 2011-02-28
Genre: Mathematics
ISBN: 3110250217

This is an essentially self-contained book on the theory of convex functions and convex optimization in Banach spaces, with a special interest in Orlicz spaces. Approximate algorithms based on the stability principles and the solution of the corresponding nonlinear equations are developed in this text. A synopsis of the geometry of Banach spaces, aspects of stability and the duality of different levels of differentiability and convexity is developed. A particular emphasis is placed on the geometrical aspects of strong solvability of a convex optimization problem: it turns out that this property is equivalent to local uniform convexity of the corresponding convex function. This treatise also provides a novel approach to the fundamental theorems of Variational Calculus based on the principle of pointwise minimization of the Lagrangian on the one hand and convexification by quadratic supplements using the classical Legendre-Ricatti equation on the other. The reader should be familiar with the concepts of mathematical analysis and linear algebra. Some awareness of the principles of measure theory will turn out to be helpful. The book is suitable for students of the second half of undergraduate studies, and it provides a rich set of material for a master course on linear and nonlinear functional analysis. Additionally it offers novel aspects at the advanced level. From the contents: Approximation and Polya Algorithms in Orlicz Spaces Convex Sets and Convex Functions Numerical Treatment of Non-linear Equations and Optimization Problems Stability and Two-stage Optimization Problems Orlicz Spaces, Orlicz Norm and Duality Differentiability and Convexity in Orlicz Spaces Variational Calculus

Optimization in Function Spaces

Optimization in Function Spaces
Author: Amol Sasane
Publisher: Courier Dover Publications
Total Pages: 260
Release: 2016-03-15
Genre: Mathematics
ISBN: 0486789454

Classroom-tested at the London School of Economics, this original, highly readable text offers numerous examples and exercises as well as detailed solutions. Prerequisites are multivariable calculus and basic linear algebra. 2015 edition.

Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces

Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces
Author: Michael Ulbrich
Publisher: SIAM
Total Pages: 315
Release: 2011-07-28
Genre: Mathematics
ISBN: 1611970687

A comprehensive treatment of semismooth Newton methods in function spaces: from their foundations to recent progress in the field. This book is appropriate for researchers and practitioners in PDE-constrained optimization, nonlinear optimization and numerical analysis, as well as engineers interested in the current theory and methods for solving variational inequalities.

Functional Analysis and Applied Optimization in Banach Spaces

Functional Analysis and Applied Optimization in Banach Spaces
Author: Fabio Botelho
Publisher: Springer
Total Pages: 584
Release: 2014-06-12
Genre: Mathematics
ISBN: 3319060740

​This book introduces the basic concepts of real and functional analysis. It presents the fundamentals of the calculus of variations, convex analysis, duality, and optimization that are necessary to develop applications to physics and engineering problems. The book includes introductory and advanced concepts in measure and integration, as well as an introduction to Sobolev spaces. The problems presented are nonlinear, with non-convex variational formulation. Notably, the primal global minima may not be attained in some situations, in which cases the solution of the dual problem corresponds to an appropriate weak cluster point of minimizing sequences for the primal one. Indeed, the dual approach more readily facilitates numerical computations for some of the selected models. While intended primarily for applied mathematicians, the text will also be of interest to engineers, physicists, and other researchers in related fields.

Optimization by Vector Space Methods

Optimization by Vector Space Methods
Author: David G. Luenberger
Publisher: John Wiley & Sons
Total Pages: 348
Release: 1997-01-23
Genre: Technology & Engineering
ISBN: 9780471181170

Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.

From Vector Spaces to Function Spaces

From Vector Spaces to Function Spaces
Author: Yutaka Yamamoto
Publisher: SIAM
Total Pages: 270
Release: 2012-10-31
Genre: Mathematics
ISBN: 1611972302

A guide to analytic methods in applied mathematics from the perspective of functional analysis, suitable for scientists, engineers and students.

Convexity and Optimization in Banach Spaces

Convexity and Optimization in Banach Spaces
Author: Viorel Barbu
Publisher: Springer Science & Business Media
Total Pages: 376
Release: 2012-01-03
Genre: Mathematics
ISBN: 940072246X

An updated and revised edition of the 1986 title Convexity and Optimization in Banach Spaces, this book provides a self-contained presentation of basic results of the theory of convex sets and functions in infinite-dimensional spaces. The main emphasis is on applications to convex optimization and convex optimal control problems in Banach spaces. A distinctive feature is a strong emphasis on the connection between theory and application. This edition has been updated to include new results pertaining to advanced concepts of subdifferential for convex functions and new duality results in convex programming. The last chapter, concerned with convex control problems, has been rewritten and completed with new research concerning boundary control systems, the dynamic programming equations in optimal control theory and periodic optimal control problems. Finally, the structure of the book has been modified to highlight the most recent progression in the field including fundamental results on the theory of infinite-dimensional convex analysis and includes helpful bibliographical notes at the end of each chapter.

Convex Optimization in Normed Spaces

Convex Optimization in Normed Spaces
Author: Juan Peypouquet
Publisher: Springer
Total Pages: 132
Release: 2015-03-18
Genre: Mathematics
ISBN: 3319137107

This work is intended to serve as a guide for graduate students and researchers who wish to get acquainted with the main theoretical and practical tools for the numerical minimization of convex functions on Hilbert spaces. Therefore, it contains the main tools that are necessary to conduct independent research on the topic. It is also a concise, easy-to-follow and self-contained textbook, which may be useful for any researcher working on related fields, as well as teachers giving graduate-level courses on the topic. It will contain a thorough revision of the extant literature including both classical and state-of-the-art references.

Convex Analysis and Optimization in Hadamard Spaces

Convex Analysis and Optimization in Hadamard Spaces
Author: Miroslav Bacak
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 217
Release: 2014-10-29
Genre: Mathematics
ISBN: 3110391082

In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject. Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics. The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.

Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization
Author: D. Butnariu
Publisher: Springer Science & Business Media
Total Pages: 218
Release: 2012-12-06
Genre: Mathematics
ISBN: 9401140669

The aim of this work is to present in a unified approach a series of results concerning totally convex functions on Banach spaces and their applications to building iterative algorithms for computing common fixed points of mea surable families of operators and optimization methods in infinite dimen sional settings. The notion of totally convex function was first studied by Butnariu, Censor and Reich [31] in the context of the space lRR because of its usefulness for establishing convergence of a Bregman projection method for finding common points of infinite families of closed convex sets. In this finite dimensional environment total convexity hardly differs from strict convexity. In fact, a function with closed domain in a finite dimensional Banach space is totally convex if and only if it is strictly convex. The relevancy of total convexity as a strengthened form of strict convexity becomes apparent when the Banach space on which the function is defined is infinite dimensional. In this case, total convexity is a property stronger than strict convexity but weaker than locally uniform convexity (see Section 1.3 below). The study of totally convex functions in infinite dimensional Banach spaces was started in [33] where it was shown that they are useful tools for extrapolating properties commonly known to belong to operators satisfying demanding contractivity requirements to classes of operators which are not even mildly nonexpansive.