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Convex Analysisの二,三の進展について

Nonlinear Programming: A Unified Approach

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Convex Analysis: (pms-28)

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Recent results in local convergence and semi-local convergence analysis constitute a natural framework for the theoretical study of iterative methods. This monograph provides a comprehensive study of both basic theory and new results in the area. Each chapter contains new theoretical results and important applications in engineering, modeling dynamic economic systems, input-output systems, optimization problems, and nonlinear and linear differential equations. Several classes of operators are considered, including operators without Lipschitz continuous derivatives, operators with high order derivatives, and analytic operators. Each section is self-contained. Examples are used to illustrate the theory and exercises are included at the end of each chapter.

The book assumes a basic background in linear algebra and numerical functional analysis. Graduate students and researchers will find this book useful. It may be used as a self-study reference or as a supplementary text for an advanced course in numerical functional analysis.

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Convergence and Applications of Newton-type Iterations

A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

The book is designed for researchers, students and practitioners interested in using fast and efficient iterative methods to approximate solutions of nonlinear equations. The following four major problems are addressed: problems 1 shows that the iterates are well defined; 2nd problem concerns the convergence of the sequences generated by a process and the question of whether the limit points are, in fact solutions of the equation; problem 3 concerns the economy of the entire operations; and last problem concerns with how to best choose a method, algorithm or software program to solve a specific type of problem and its description of when a given algorithm succeeds or fails.The book contains applications in several areas of applied sciences including mathematical programming and mathematical economics. There is also a huge number of exercises complementing the theory. This book contains the latest convergence results for the iterative methods: Iterative methods with the least computational cost; Iterative methods with the weakest convergence conditions; and Open problems on iterative methods.

Computational Theory of Iterative Methods

Weaker conditions for the convergence of Newton's method

A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced. The results are then applied to solve Chandrasekhar's integral equation and related equations without the usual positivity assumptions.

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Ioannis K. Argyros

Papers

Convergence and Applications of Newton-type Iterations

A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

Computational Theory of Iterative Methods

Weaker conditions for the convergence of Newton's method

Quadratic equations and applications to Chandrasekhar's and related equations