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Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization




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ISBN 9781439868201
Published July 18, 2013 by Chapman and Hall/CRC
280 Pages 17 B/W Illustrations

 
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Book Description

Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction.

The first part of the book focuses on generalized convexity and generalized monotonicity. The authors investigate convexity and generalized convexity for both the differentiable and nondifferentiable case. For the nondifferentiable case, they introduce the concepts in terms of a bifunction and the Clarke subdifferential.

The second part offers insight into variational inequalities and optimization problems in smooth as well as nonsmooth settings. The book discusses existence and uniqueness criteria for a variational inequality, the gap function associated with it, and numerical methods to solve it. It also examines characterizations of a solution set of an optimization problem and explores variational inequalities defined by a bifunction and set-valued version given in terms of the Clarke subdifferential.

Integrating results on convexity, monotonicity, and variational inequalities into one unified source, this book deepens your understanding of various classes of problems, such as systems of nonlinear equations, optimization problems, complementarity problems, and fixed-point problems. The book shows how variational inequality theory not only serves as a tool for formulating a variety of equilibrium problems, but also provides algorithms for computational purposes.

Table of Contents

Generalized Convexity and Generalized Monotonicity
Elements of Convex Analysis
Preliminaries and Basic Concepts
Convex Sets
Hyperplanes
Convex Functions
Generalized Convex Functions
Optimality Criteria
Subgradients and Subdifferentials

Generalized Derivatives and Generalized Subdifferentials
Directional Derivatives
Gâteaux Derivatives
Dini and Dini-Hadamard Derivatives
Clarke and Other Types of Derivatives
Dini and Clarke Subdifferentials

Nonsmooth Convexity
Nonsmooth Convexity in Terms of Bifunctions
Generalized Nonsmooth Convexity in Terms of Bifunctions
Generalized Nonsmooth Convexity in Terms of Subdifferentials
Generalized Nonsmooth Pseudolinearity in Terms of Clarke Subdifferentials

Monotonocity and Generalized Monotonicity
Monotonicity and Its Relation with Convexity
Nonsmooth Monotonicity and Generalized Monotonicity in Terms of a Bifunction
Relation between Nonsmooth Monotonicity and Nonsmooth Convexity
Nonsmooth Pseudoaffine Bifunctions and Nonsmooth Pseudolinearity
Generalized Monotonicity for Set-Valued Maps

Nonsmooth Variational Inequalities and Nonsmooth Optimization
Elements of Variational Inequalities
Variational Inequalities and Related Problems
Basic Existence and Uniqueness Results
Gap Functions
Solution Methods

Nonsmooth Variational Inequalities
Nonsmooth Variational Inequalities in Terms of a Bifunction
Relation between an Optimization Problem and Nonsmooth Variational Inequalities
Existence Criteria
Extended Nonsmooth Variational Inequalities
Gap Functions and Saddle Point Characterization

Characterizations of Solution Sets of Optimization Problem and Nonsmooth Variational Inequalities
Characterizations of the Solution Set of an Optimization Problem with a Pseudolinear Objective Function
Characterizations of the Solution Set of Variational Inequalities Involving Pseudoaffine Bifunctions
Lagrange Multiplier Characterizations of Solution Set of an Optimization Problem

Nonsmooth Generalized Variational Inequalities and Optimization Problems
Generalized Variational Inequalities and Related Topics
Basic Existence and Uniqueness Results
Gap Functions for Generalized Variational Inequalities
Generalized Variational Inequalities in Terms of the Clarke Subdifferential and Optimization Problems
Characterizations of Solution Sets of an Optimization Problem with Generalized Pseudolinear Objective Function

Appendix A: Set-Valued Maps
Appendix B: Elements of Nonlinear Analysis

Index

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Reviews

"Overall, the book contains a lot of interesting material on the generalized convexity and generalized monotonicity with important applications to variational inequalities in finite dimensions. The book is nicely written with good examples and figures, making it useful also for advanced undergraduate students."
—B. Mordukhovich, Mathematical Reviews, January 2014