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Fixed Point Theory, Variational Analysis, and Optimization

Saleh Abdullah R. Al-Mezel, Falleh Rajallah M. Al-Solamy, andQamrul Hasan Ansari, editors
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2014
Number of Pages: 
347
Format: 
Hardcover
Price: 
129.95
ISBN: 
9781482222074
Category: 
Proceedings
We do not plan to review this book.

Preface

List of Figures

List of Tables

Contributors

 

I. Fixed Point Theory

 

Common Fixed Points in Convex Metric Spaces

Abdul Rahim Khan and Hafiz Fukhar-ud-din

Introduction

Preliminaries

Ishikawa Iterative Scheme

Multistep Iterative Scheme

One-Step Implicit Iterative Scheme

Bibliography

 

Fixed Points of Nonlinear Semigroups in Modular Function Spaces

B. A. Bin Dehaish and M. A. Khamsi

Introduction

Basic Definitions and Properties

Some Geometric Properties of Modular Function Spaces

Some Fixed-Point Theorems in Modular Spaces

Semigroups in Modular Function Spaces

Fixed Points of Semigroup of Mappings

Bibliography

 

Approximation and Selection Methods for Set-Valued Maps and Fixed Point Theory

Hichem Ben-El-Mechaiekh

Introduction

Approximative Neighborhood Retracts, Extensors, and Space Approximation

Approximative Neighborhood Retracts and Extensors

Contractibility and Connectedness

Contractible Spaces

Proximal Connectedness

Convexity Structures

Space Approximation

The Property A(K;P) for Spaces

Domination of Domain

Domination, Extension, and Approximation

Set-Valued Maps, Continuous Selections, and Approximations

Semicontinuity Concepts

USC Approachable Maps and Their Properties

Conservation of Approachability

Homotopy Approximation, Domination of Domain, and Approachability

Examples of A−Maps

Continuous Selections for LSC Maps

Michael Selections

A Hybrid Continuous Approximation-Selection Property

More on Continuous Selections for Non-Convex Maps

Non-Expansive Selections

Fixed Point and Coincidence Theorems

Generalizations of the Himmelberg Theorem to the Non-Convex Setting

Preservation of the FPP from P to A(K;P)

A Leray-Schauder Alternative for Approachable Maps

Coincidence Theorems

Bibliography

 

II. Convex Analysis and Variational Analysis

 

Convexity, Generalized Convexity, and Applications

N. Hadjisavvas

Introduction

Preliminaries

Convex Functions

Quasiconvex Functions

Pseudoconvex Functions

On the Minima of Generalized Convex Functions

Applications

Sufficiency of the KKT Conditions

Applications in Economics

Further Reading

Bibliography

 

New Developments in Quasiconvex Optimization

D. Aussel

Introduction

Notations

The Class of Quasiconvex Functions

Continuity Properties of Quasiconvex Functions

Differentiability Properties of Quasiconvex Functions

Associated Monotonicities

Normal Operator: A Natural Tool for Quasiconvex Functions

The Semistrictly Quasiconvex Case

The Adjusted Sublevel Set and Adjusted Normal Operator

Adjusted Normal Operator: Definitions

Some Properties of the Adjusted Normal Operator

Optimality Conditions for Quasiconvex Programming

Stampacchia Variational Inequalities

Existence Results: The Finite Dimensions Case

Existence Results: The Infinite Dimensional Case

Existence Result for Quasiconvex Programming

Bibliography

 

An Introduction to Variational-Like Inequalities

Qamrul Hasan Ansari

Introduction

Formulations of Variational-Like Inequalities

Variational-Like Inequalities and Optimization Problems

Invexity

Relations between Variational-Like Inequalities and an Optimization Problem

Existence Theory

Solution Methods

Auxiliary Principle Method

Proximal Method

Appendix

Bibliography

 

III. Vector Optimization

 

Vector Optimization: Basic Concepts and Solution Methods

Dinh The Luc and Augusta Ratiu

Introduction

Mathematical Backgrounds

Partial Orders

Increasing Sequences

Monotone Functions

Biggest Weakly Monotone Functions

Pareto Maximality

Maximality with Respect to Extended Orders

Maximality of Sections

Proper Maximality and Weak Maximality

Maximal Points of Free Disposal Hulls

Existence

The Main Theorems

Generalization to Order-Complete Sets

Existence via Monotone Functions

Vector Optimization Problems

Scalarization

Optimality Conditions

Differentiable Problems

Lipschitz Continuous Problems

Concave Problems

Solution Methods

Weighting Method

Constraint Method

Outer Approximation Method

Bibliography

 

Multi-Objective Combinatorial Optimization

Matthias Ehrgott and Xavier Gandibleux

Introduction

Definitions and Properties

Two Easy Problems: Multi-Objective Shortest Path and Spanning Tree

Nice Problems: The Two-Phase Method

The Two-Phase Method for Two Objectives

The Two-Phase Method for Three Objectives

Difficult Problems: Scalarization and Branch and Bound

Scalarization

Multi-Objective Branch and Bound

Challenging Problems: Metaheuristics

Conclusion

Bibliography

 

Index

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