Quadratic Programming and Affine Variational Inequalities: A Qualitative Study

Gue Myung Lee, Nguyen Nang Tam, Nguyen Dong Yen (auth.)9780387242774, 0-387-24277-5

This book develops a unified theory on qualitative aspects of nonconvex quadratic programming and affine variational inequalities. The first seven chapters introduce the reader step-by-step to the central issues concerning a quadratic program or an affine variational inequality, such as the solution existence, necessary and sufficient conditions for a point to belong to the solution set, and properties of the solution set. The subsequent two chapters briefly discuss two concrete models (a linear fractional vector optimization and a traffic equilibrium problem) whose analysis can benefit greatly from using the results on quadratic programs and affine variational inequalities. There are six chapters devoted to the study of continuity and differentiability properties of the characteristic maps and functions in quadratic programs and in affine variational inequalities where all the components of the problem data are subject to perturbation. Quadratic programs and affine variational inequalities under linear perturbations are studied in three other chapters.

One special feature of this book is that when a certain property of a characteristic map or function is investigated, the authors always try first to establish necessary conditions for it to hold, then they go on to study whether the obtained necessary conditions are sufficient ones. This helps to clarify the structures of the two classes of problems under consideration. The qualitative results can be used for dealing with algorithms and applications related to quadratic programming problems and affine variational inequalities.

Audience

This book is intended for graduate and postgraduate students in applied mathematics, as well as researchers in the fields of nonlinear programming and equilibrium problems. It can be used for some advanced courses on nonconvex quadratic programming and affine variational inequalities.

Table of contents :
Quadratic Programming Problems….Pages 1-28
Existence Theorems for Quadratic Programs….Pages 29-44
Necessary and Sufficient Optimality Conditions for Quadratic Programs….Pages 45-63
Properties of the Solution Sets of Quadratic Programs….Pages 65-84
Affine Variational Inequalities….Pages 85-102
Solution Existence for Affine Variational Inequalities….Pages 103-118
Upper-Lipschitz Continuity of the Solution Map in Affine Variational Inequalities….Pages 119-142
Linear Fractional Vector Optimization Problems….Pages 143-154
The Traffic Equilibrium Problem….Pages 155-162
Upper Semicontinuity of the KKT Point Set Mapping….Pages 163-194
Lower Semicontinuity of the KKT Point Set Mapping….Pages 195-211
Continuity of the Solution Map in Quadratic Programming….Pages 213-222
Continuity of the Optimal Value Function in Quadratic Programming….Pages 223-238
Directional Differentiability of the Optimal Value Function….Pages 239-257
Quadratic Programming under Linear Perturbations: I. Continuity of the Solution Maps….Pages 259-268
Quadratic Programming under Linear Perturbations: II. Properties of the Optimal Value Function….Pages 269-290
Quadratic Programming under Linear Perturbations: III. The Convex Case….Pages 291-305
Continuity of the Solution Map in Affine Variational Inequalities….Pages 307-327