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2002 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Complementarity, Equilibrium, Efficiency and Economics

verfasst von: G. Isac, V. A. Bulavsky, V. V. Kalashnikov

Verlag: Springer US

Buchreihe : Nonconvex Optimization and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

In complementarity theory, which is a relatively new domain of applied mathematics, several kinds of mathematical models and problems related to the study of equilibrium are considered from the point of view of physics as well as economics. In this book the authors have combined complementarity theory, equilibrium of economical systems, and efficiency in Pareto's sense. The authors discuss the use of complementarity theory in the study of equilibrium of economic systems and present results they have obtained. In addition the authors present several new results in complementarity theory and several numerical methods for solving complementarity problems associated with the study of economic equilibrium. The most important notions of Pareto efficiency are also presented.
Audience: Researchers and graduate students interested in complementarity theory, in economics, in optimization, and in applied mathematics.

Inhaltsverzeichnis

Frontmatter

Economic Models and Complementarity

Frontmatter
Chapter 1. Introduction
Abstract
To a certain degree, the concept of complementarity is analogous to the concept of a stationary point in the extremum problems.If the point z̄ is a (local) minimum of a real differentiable function / defined over the positive half-axis R + = [0,+∞] then the inequality f’(0) ≥ 0 is the necessary condition of that.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 2. Optimization Models
Abstract
This section deals mainly with the problem of minimization of the function f : D → R over the closed convex subset K ⊂ D. The domain D ⊂ R n , in general, may not coincide with the whole space R n . However, in order to avoid considering the boundary effects, we will always assume that D is and open set. Thus, the closed subset K is contained in the interior of D. It is traditional to consider two classes of these problems: the case of continuously diffjerentiable function f and the case of convex function f.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 3. General Economic Equilibrium
Abstract
The multi-objective models analyzed in Section 2.4 and, partially, in Section 2.3, dealt mainly with the case of a single decision-maker. Even when we supposed that each criterion described the aim of a distinct subject, these aims were uniform enough. Otherwise, the selection of efficient points and the scalarization methods could be strongly criticized. In this and iij the next sections of Chapter 3, we will consider a thoroughly different situation.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 4. Models of Oligopoly
Abstract
As an example of an important complementarity problem, a model of an oligopolistic market with a homogeneous product is examined in this chapter. In Section 4.2, a generalized Cournot model is introduced. Each subject of the model uses a conjecture about the market response to variations of its production volume. The conjecture value depends upon both the current total volume of production at the market and the subject’s contribution into it. Under general enough assumptions, the equilibrium existence and uniqueness theorems are proven. When analyzing the network equilibrium model, the topological degree theory is used to prove the existence theorem.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 5. Oligopoly with Leaders
Abstract
This chapter is dedicated to the Stackelberg model which is a particular case of bilevel problems with equilibrium constraints. In Section 5.1, the Stackelberg model is extended to the case of several leaders, and the theorem of existence of a stationary point is obtained. In Section 5.2, we compare the equilibria in the Stackelberg and Cournot models. Section 5.3 presents simple examples of comparison of equilibria in different models: Cournot model, high expectations model, Stackelberg model, and the perfect competition one. These examples illustrate results of Section 5.2. At last, problems of efficient computation of the equilibrium are considered in Section 5.4.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov

General Complementarity Problems

Frontmatter
Chapter 6. Complementarity Problems with Respect to General Cones
Abstract
The topological degree theory is a powerful tool to study the problem of existence of solutions to nonlinear equations. Since recently, this theory has been widely used in applications to a more general problem, namely, the complementarity problem (CP). In this chapter, we consider both the complementarity problem with respect to an arbitrary cone of the Euclidean space, and its variations: the standard CP (with the nonnegative orthant R + n as the cone), the implicit CP, the general order CP, and the semideflnite CP defined for the real matrices. Having specified the results of Chapter 1 of this book (Proposition 1.5, Proposition 1.6) and definitions of exceptional family of elements for the particular types of complementarity problems, we obtain existence theorems.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 7. Pseudomonotone and Implicit Complementarity Problems
Abstract
In this chapter, we extend our techniques to the case of infinite qimensional complementarity problems. Especial attention is paid to the latter with pseu-domonotone operators. The second part of the chapter is devoted to Implicit Complementarity Problems with single-valued and multi-valued mappings.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov

Numerical Methods for Solving Complementarity Problems

Frontmatter
Chapter 8. Complementarity Pivot Methods
Abstract
This chapter deals with pivot methods that constituted the first and important instrument in solving Linear Complementarity Problems. Developed under the impact of the simplex method, they revealed many crucial differences between the linear programming and complementarity problems.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 9. Scarf Type Algorithms
Abstract
Citing Scarf (1973), one of the major triumphs of mathematical economics has been the proof of the existence of a solution for the neoclassic model of economic equilibrium. When cast in a mathematical form the general equilibrium model becomes a system of simultaneous equations and inequalities so complex that the existence of a solution can be guaranteed only by an appeal to fixed point theorems rather than by more elementary and constructively oriented techniques.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 10. Newton-Like Methods
Abstract
This chapter deals with the problem of finding out an equilibrium in an oligopolistic market model where several subjects supply a single homogeneous product in a non-cooperative manner. The problem is reduced to a nonlinear equation some terms of which are determined by solving nonlinear complementarity problems. An algorithm is presented that combines the Newton method steps with the dichotomy techniques. Under certain assumptions, the algorithm is shown to be convergent at the quadratic rate. Finally, the algorithm is extended to the case of nonlinear production costs, and its linear convergence is demonstrated.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 11. Parametrization and Reduction to Nonlinear Equations
Abstract
This chapter is dedicated to two different types of methods solving nonlinear complementarity problems. The first one is a method of approximate solution of nonlinear complementarity problem with parameters: Given a continuous mapping f : R n × R m → R n , and a fixed vector of parameters u = (u 1, ..., u m ) T , find a x ∈ R n such that
$$ x \ge 0,\quad f(x,u) \ge 0,\quad and\quad {x^T}f(x,u) = 0 $$
(11.1)
.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov

Efficiency in Abstract Spaces

Frontmatter
Chapter 12. Efficiency
Abstract
A very popular domain of applied mathematics is optimization, because the diversity of its applications to economics, engineering and sciences. Certainly the applications to practical problems stimulated the impressive development of this domain. Between the chapters of optimization, one is the optimization of vector-valued functions, known also under the name of Pareto optimization. In 1906 V. Pareto wrote: “Principeremo con deftnire un termine di cui è comodo fare uso per scansare lungaggini. Diremo che i componenti di una colletivita godono, in una certa postione, del massimo di ofelimita, quando è impossibile allontanarsi pochissimo da quella positione giovando, o nuocendo, a tutti i componenti la collectività; ogni picolissimo spostamento da quella positione avendo necessariamente per effetto di giovare a parte dei componenti la collectività e di nuocere ad altri.” (Pareto, 1919).
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Chapter 13. Approximative Efficiency
Abstract
A new direction in the study of efficiency, which is now in developing, is the study of ε-efficiency or of other forms of approximative efficiency. In particular, the ε-efficiency is related to the study of ε-solutions in vector optimization problems. Begining with the paper of Loridan (1984) several concepts for approximately efficient solutions of a vector optimization problem were published in the last years. We mention the works by Németh (1986), Staib (1988), Valyi (1985), Gerth (Tammer) (1978), Tammer (1992), Tammer [l]-[3] (1993), Helbig e.a. (1992), Isac (1984, 1986), Gopfert and Tammer [1],[3] (1995), Göpfert and Tammer (1998) among others. This chapter is dedicated to the study of approximative efficiency.
G. Isac, V. A. Bulavsky, V. V. Kalashnikov
Backmatter
Metadaten
Titel
Complementarity, Equilibrium, Efficiency and Economics
verfasst von
G. Isac
V. A. Bulavsky
V. V. Kalashnikov
Copyright-Jahr
2002
Verlag
Springer US
Electronic ISBN
978-1-4757-3623-6
Print ISBN
978-1-4419-5223-3
DOI
https://doi.org/10.1007/978-1-4757-3623-6