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