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Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization

By Qamrul Hasan Ansari, C. S. Lalitha, Monika Mehta

Chapman and Hall/CRC – 2013 – 280 pages

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    978-1-43-986820-1
    July 18th 2013

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.

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

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

Name: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization (Hardback)Chapman and Hall/CRC 
Description: By Qamrul Hasan Ansari, C. S. Lalitha, Monika Mehta. 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...
Categories: Operations Research, Operations Research, Mathematics & Statistics for Engineers