1st Edition
Fixed Point Theory, Variational Analysis, and Optimization
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