1st Edition

Fixed Point Theory, Variational Analysis, and Optimization

ISBN 9781482222074
Published June 3, 2014 by Chapman and Hall/CRC
368 Pages 42 B/W Illustrations

USD $200.00

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Book Description

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.

Table of Contents


List of Figures

List of Tables


I. Fixed Point Theory

Common Fixed Points in Convex Metric Spaces

Abdul Rahim Khan and Hafiz Fukhar-ud-din



Ishikawa Iterative Scheme

Multistep Iterative Scheme

One-Step Implicit Iterative Scheme


Fixed Points of Nonlinear Semigroups in Modular Function Spaces

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


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


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

Hichem Ben-El-Mechaiekh


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


II. Convex Analysis and Variational Analysis

Convexity, Generalized Convexity, and Applications

N. Hadjisavvas



Convex Functions

Quasiconvex Functions

Pseudoconvex Functions

On the Minima of Generalized Convex Functions


Sufficiency of the KKT Conditions

Applications in Economics

Further Reading


New Developments in Quasiconvex Optimization

D. Aussel



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


An Introduction to Variational-Like Inequalities

Qamrul Hasan Ansari


Formulations of Variational-Like Inequalities

Variational-Like Inequalities and Optimization Problems


Relations between Variational-Like Inequalities and an Optimization Problem

Existence Theory

Solution Methods

Auxiliary Principle Method

Proximal Method



III. Vector Optimization

Vector Optimization: Basic Concepts and Solution Methods

Dinh The Luc and Augusta Ratiu


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


The Main Theorems

Generalization to Order-Complete Sets

Existence via Monotone Functions

Vector Optimization Problems


Optimality Conditions

Differentiable Problems

Lipschitz Continuous Problems

Concave Problems

Solution Methods

Weighting Method

Constraint Method

Outer Approximation Method


Multi-Objective Combinatorial Optimization

Matthias Ehrgott and Xavier Gandibleux


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


Multi-Objective Branch and Bound

Challenging Problems: Metaheuristics




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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