Quadratic Programming and Affine Variational Inequalities: A Qualitative Study

ISBN-13: 9780387242774 / Angielski / Twarda / 2005 / 346 str.

Gue M. Lee; Nguyen N. Tam; Nguyen D. Yen

Opis

Quadratic programs and affine variational inequalities represent two fundamental, closely-related classes of problems in the t, heories of mathematical programming and variational inequalities, resp- tively. This book develops a unified theory on qualitative aspects of nonconvex quadratic programming and affine variational inequ- ities. 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 discuss briefly two concrete nlodels (linear fractional vector optimization and the traffic equilibrium problem) whose analysis can benefit a lot from using the results on quadratic programs and affine variational inequalities. There are six chapters devoted to the study of conti- ity and/or differentiability properties of the characteristic maps and functions in quadratic programs and in affine variational inequa- ties where all the components of the problem data are subject to perturbation. Quadratic programs and affine variational inequa- ties under linear perturbations are studied in three other chapters. One special feature of the presentation is that when a certain pr- erty of a characteristic map or function is investigated, we always try first to establish necessary conditions for it to hold, then we go on to study whether the obtained necessary conditions are suf- cient ones. This helps to clarify the structures of the two classes of problems under consideration

Recenzje

From the reviews:

"This book presents a detailed exposition of qualitative results for quadratic programming (QP) and affine variational inequalities (AVI). Both topics are developed into a unifying approach." (Walter Gómez Bofill, Zentralblatt MATH, Vol. 1092 (18), 2006)

"This book presents a theory of qualitative aspects of nonconvex quadratic programs and affine variational inequalities. ... Applications to fractional vector optimization problems and traffic equilibrium problems are discussed, too. The book is a valuable collection of many basic ideas and results for these classes of problems, and it may be recommended to researchers and advanced students not only in the field of optimization, but also in other fields of applied mathematics." (D. Klatte, Mathematical Reviews, Issue 2006 e)

"This book presents a qualitative study of nonconvex quadratic programs and affine variational inequalities. ... Most of the proofs are presented in a detailed and elementary way. ... Whenever possible, the authors give examples illustrating their results. ... In summary, this book can be recommended for advanced students in applied mathematics due to the clear and elementary style of presentation. ... this book can be serve as an interesting reference for researchers in the field of quadratic programming, finite dimensional variational inequalities and complementarity problems." (M. Stingl, Mathemataical Methods of Operations Research, Vol. 65, 2007)

Spis treści

Preface—Notations and Abbreviations—1. Quadratic Programming Problems—2. Existence Theorems for Quadratic Programs—3. Necessary and Sufficient Optimality Conditions for Quadratic Programs—4. Properties of the Solution Sets of Quadratic Programs—5. Affine Variational Inequalities—6. Solution Existence for Affine Variational Inequalities—7. Upper-Lipschitz Continuity of the Solution Map in Affine Variational Inequalities—8. Linear Fractional Vector Optimization Problems—9. The Traffic Equilibrium Problem—10. Upper Semicontinuity of the KKT Point Set Mapping—11. Lower Semicontinuity of the KKT Point Set Mapping—12. Continuity of the Solution Map in Quadratic Programming—13. Continuity of the Optimal Value Function in Quadratic Programming—14. Directional Differentiability of the Optimal Value Function—15. Quadratic Programming Under Linear Perturbations: I. Continuity of the Solution Maps—16. Quadratic Programming Under Linear Perturbations: II. Properties of the Optimal Value Function—17. Quadratic Programming Under Linear Perturbations: III. The Convex Case—18. Continuity of the Solution Map in Affine Variational Inequalities—References—Index

Blurb

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.