Fixed point theory, variational analysis, and optimization / edited by Saleh A. R. Al-Mezel, Falleh R. M. Al-Solamy, Qamrul H. Ansari.

Contributor(s):

Material type: Text

TextPublication details: FL : CRC Press, c2014.Description: xx, 347 pages : illustrations; 25 cmISBN:

9781482222074 (hardcover : acidfree paper)
1482222078 (hardcover : acidfree paper)

Subject(s):

DDC classification:

514.74 23 AL-M

LOC classification:

QA427 .A46 2014

Contents:

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

Summary: Fixed Point Theory, Variational Analysis, and Optimization not only covers three vital branches of nonlinear analysis—fixed point theory, variational inequalities, and vector optimization—but also explains the connections between them, enabling the study of a general form of variational inequality problems related to the optimality conditions involving differentiable or directionally differentiable functions. This essential reference supplies both an introduction to the field and a guideline to the literature, progressing from basic concepts to the latest developments. Packed with detailed proofs and bibliographies for further reading, the text: Examines Mann-type iterations for nonlinear mappings on some classes of a metric space Outlines recent research in fixed point theory in modular function spaces Discusses key results on the existence of continuous approximations and selections for set-valued maps with an emphasis on the nonconvex case Contains definitions, properties, and characterizations of convex, quasiconvex, and pseudoconvex functions, and of their strict counterparts Discusses variational inequalities and variational-like inequalities and their applications Gives an introduction to multi-objective optimization and optimality conditions Explores multi-objective combinatorial optimization (MOCO) problems, or integer programs with multiple objectives Fixed Point Theory, Variational Analysis, and Optimization is a beneficial resource for the research and study of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics. It provides fundamental knowledge of directional derivatives and monotonicity required in understanding and solving variational inequality problems.

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Holdings
Item type	Current library	Collection	Call number	Status	Date due	Barcode
Project book	CUTN Central Library Sciences	Non-fiction	514.74 AL-M (Browse shelf(Opens below))	Checked out to Renuka Devi V (20019T)	31/01/2024	48827

"A Chapman & Hall Book."

Includes bibliographical references (pages 334-341) and index.

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

Fixed Point Theory, Variational Analysis, and Optimization not only covers three vital branches of nonlinear analysis—fixed point theory, variational inequalities, and vector optimization—but also explains the connections between them, enabling the study of a general form of variational inequality problems related to the optimality conditions involving differentiable or directionally differentiable functions. This essential reference supplies both an introduction to the field and a guideline to the literature, progressing from basic concepts to the latest developments. Packed with detailed proofs and bibliographies for further reading, the text:

Examines Mann-type iterations for nonlinear mappings on some classes of a metric space
Outlines recent research in fixed point theory in modular function spaces
Discusses key results on the existence of continuous approximations and selections for set-valued maps with an emphasis on the nonconvex case
Contains definitions, properties, and characterizations of convex, quasiconvex, and pseudoconvex functions, and of their strict counterparts
Discusses variational inequalities and variational-like inequalities and their applications
Gives an introduction to multi-objective optimization and optimality conditions
Explores multi-objective combinatorial optimization (MOCO) problems, or integer programs with multiple objectives
Fixed Point Theory, Variational Analysis, and Optimization is a beneficial resource for the research and study of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics. It provides fundamental knowledge of directional derivatives and monotonicity required in understanding and solving variational inequality problems.

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