Vector Variational Inequalities and Vector Optimization

Qamrul Hasan Ansari is Professor of Mathematics at Aligarh Muslim University, India. His research interest lies in applied functional analysis, optimization, convex analysis, nonlinear analysis, fixed point theory in topological vector spaces, abstract economies and game theory. He looks back at 28 years of teaching and research experience and is co-author of two further books and editor/co-editor of seven books.

Elisabeth Köbis is lecturer and researcher at Martin-Luther-University Halle-Wittenberg, Germany. She received her PhD from Martin-Luther-University Halle-Wittenberg, Germany. Her research interest lies in set optimization, vector optimization, robust and uncertain optimization, robust approaches to uncertain multi-objective optimization problems and unified approaches to uncertain optimization using nonlinear scalarization.

This book presents the mathematical theory of vector variational inequalities and their relations with vector optimization problems. It is the first-ever book to introduce well-posedness and sensitivity analysis for vector equilibrium problems. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, set-valued analysis and fixed-point theory for set-valued maps, as well as a brief introduction to variational inequalities and equilibrium problems. Chapter 2 presents an overview of analysis over cones, including continuity and convexity of vector-valued functions. The book then shifts its focus to solution concepts and classical methods in vector optimization. It describes the formulation of vector variational inequalities and their applications to vector optimization, followed by separate chapters on linear scalarization, nonsmooth and generalized vector variational inequalities. Lastly, the book introduces readers to vector equilibrium problems and generalized vector equilibrium problems. Written in an illustrative and reader-friendly way, the book offers a valuable resource for all researchers whose work involves optimization and vector optimization.

Presents for the first time a mathematical theory on vector variational inequalities and vector equilibrium problems

Preliminaries.- Analysis over Cones.- Solution Concepts in Vector Optimization.- Classical Methods in Vector Optimization.- Vector Variational Inequalities.- Linear Scalarization of Vector Variational Inequalities.- Nonsmooth Vector Variational Inequalities.- Generalized Vector Variational Inequalities.- Vector Equilibrium Problems.- Generalized Vector Equilibrium Problems.