1st Edition

Fixed Point Theory, Variational Analysis, and Optimization

    368 Pages 42 B/W Illustrations
    by Chapman & Hall

    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.

    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

    Biography

    Saleh Abdullah R. Al-Mezel is a full professor of mathematics at King Abdulaziz University, Jeddah, Saudi Arabia and the vice president for academic affairs at the University of Tabuk, Saudi Arabia. He holds a B.Sc from King Abdulaziz University; an M.Phil from Swansea University, Wales; and a Ph.D from Cardiff University, Wales. He possesses over ten years of teaching experience and has participated in several sponsored research projects. His publications span numerous books and international journals.

    Falleh Rajallah M. Al-Solamy is a professor of mathematics at King Abdulaziz University, Jeddah, Saudi Arabia and the vice president for graduate studies and scientific research at the University of Tabuk, Saudi Arabia. He holds a B.Sc from King Abdulaziz University and a Ph.D from Swansea University, Wales. A member of several academic societies, he possesses over 7 years of academic and administrative experience. He has completed 30 research projects on differential geometry and its applications, participated in over 14 international conferences, and published more than 60 refereed papers.

    Qamrul Hasan Ansari is a professor of mathematics at Aligarh Muslim University, India, from which he also received his M.Phil and Ph.D. He has co/edited, co/authored, and/or contributed to 8 scholarly books. He serves as associate editor of the Journal of Optimization Theory and Applications and the Fixed Point Theory and Applications, and has guest-edited special issues of several other journals. He has more than 150 research papers published in world-class journals and his work has been cited in over 1,400 ISI journals. His fields of specialization and/or interest include nonlinear analysis, optimization, convex analysis, and set-valued analysis.

    "There is a real need for this book. It is useful for people who work in areas of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics."
    —Nan-Jing Huang, Sichuan University, Chengdu, People’s Republic of China