1st Edition

Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization

    296 Pages 17 B/W Illustrations
    by Chapman & Hall

    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.

    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

    Biography

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

    "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