New Book: Equilibrium Models and Variational Inequalities

by Igor Konnov (ikonnov@ksu.ru)

(Mathematics in Science and Engineering, Vol.210), ISBN 0-444-53030-4, 248 pp., 2007.
Author: Igor Konnov, Kazan University
Publisher: Elsevier B.V.

Features of book:
The concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics),
economics, engineering, transportation, sociology, chemistry, biology and other fields. If one can formulate the
equilibrium problem in the form of a mathematical model, solutions of the corresponding problem can be used
for forecasting the future behavior of very complex systems and, also, for correcting the the current
state of the system under control. It is very essential to note that the concept of equilibrium is
not restricted by the static problems only, hence, one can consider the equilibrium trajectories
which correspond to dynamic equilibria. This book can be regarded as an attempt to present
a unifying look on different equilibrium concepts in economics, which is based on the variational
inequality approach. In order to illustrate this approach, we include several models from related sciences.
The emphasis is made on revealing generic properties in different models. The book describes also the present
state of investigations in this field.
The first part is devoted to known classes of models which illustrate the diversity of concepts of equilibrium
in complex systems. It includes, in particular, static and dynamic input-output models, Walras, Cassel-Wald,
spatial price, and oligopolistic equilibrium models, moreover, transportation and migration equilibrium models.
In the second part, we consider the basics of theory and solution methods for one of the most popular formulation
of equilibrium known as the complementarity problem. This includes existence and uniqueness of solutions, stability
of equilibria, existence of natural mechanisms for attaining equilibrium states, and creating effective algorithms
for finding equilibrium solutions under order monotonicity assumptions.
In the third part, we consider similar issues in the theory and solution methods for variational inequality problems
under general monotonicity assumptions and discuss their applications to the above models. The exposition is rather
elementary and destined for students, lecturers and specialists both in Economics and in Applied Mathematics.
Many exercises are included. The book is based on author's courses given in several universities mainly
in Italy, Finland, and Russia.

Contents:
Preface
1 Introduction

Part I : Models
2 Linear Models in Economics
3 Linear Dynamic Models
4 Optimization and Equilibria
5 Nonlinear Economic Equilibrium Models
6 Transportation and Migration Equilibrium Models

Part II : Complementarity Problems
7 Complementarity Problems with Z Properties
8 Applications
9 Complementarity Problems with P Properties
10 Applications

Part III: Variational Inequalities
11 Theory of Variational Inequalities
12 Applications
13 Projection Type Methods and Their Extensions
14 Applications of the Projection Methods
15 Regularization Methods
16 Direct Iterative Methods for Monotone Variational Inequalities
17 Solutions to Exercises
References
Index

Homepage: http://www.elsevier.com/wps/find/bookdescription.cws_home/710630/description#description