Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization

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