Economic Equilibrium Problems Research Articles

Existence and Approximation of Robust Solutions of Variational Inequality Problems over Polytopes

We study nonlinear variational inequality problems over polytopes from a viewpoint of stability and propose a new solution concept. Extending an earlier concept proposed by Yang [Z. Yang, SIAM J. Control Optim., 34 (1996), pp. 491--506] on the unit simplex, we will introduce the concept of the robust stationary point, which is a refinement of the concept of the stationary point. Though a stationary point need not be robust, it is shown that every continuous function on a polytope has a robust stationary point. We develop a simplicial algorithm to compute a robust stationary point of a continuous function on a polytope. The algorithm can be briefly stated as follows. Starting with any point in the relative interior of a polytope, the algorithm generates a piecewise linear path which leads to an approximate robust stationary point of any a priori chosen accuracy within a finite number of steps. Moreover, we also discuss several numerical examples and apply the new concept to noncooperative games and economic equilibrium problems.

An Introduction to Simulated Annealing Algorithms for the Computation ofEconomic Equilibrium

Economic equilibrium computation has raised the issue of global optimization algorithms since economic equilibrium problems can be cast as a global optimization problem. However, nearly all conventional algorithms stop when they find a local optimum. Over the last decade a number of new optimization algorithms have appeared, simulated annealing is one of them. It is a powerful stochastic search algorithm applicable to a wide range of problems for which little prior knowledge is available, and it asymptotically probabilistically converges to a global optimum. In this paper, we will give a brief introduction to simulated annealing and apply it to the computation of economic equilibrium. We also reported our computational experience in the paper. This early result shows that the application of simulated annealing to computation of economic equilibrium is encouraging and it deserves further research.

A reduction method for variational inequalities

This paper explains a method by which the number of variables in a variational inequality having a certain form can be substantially reduced by changing the set over which the variational inequality is posed. The method applies in particular to certain economic equilibrium problems occurring in applications. We explain and justify the method, and give examples of its application, including a numerical example in which the solution time for the reduced problem was approximately 2% of that for the problem in its original form. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Computational experience with general equilibrium problems

We report on computational experience with an implementation of three algorithms for the general economic equilibrium problem. As a result we get that the projection algorithm for variational inequalities increases the size of solvable models by a factor of 5–10 in comparison with the classical homotopy method. As a third approach we implemented a simulated annealing heuristic which might be suitable to estimate equilibria for very large models.

General economic equilibrium and variational inequalities: existence, uniqueness and sensitivity

In this paper we use the theory of variational inequalities to study the general economic equilibrium problem with production. The variational inequality is defined on a bounded set. Existence, uniqueness and sensitivity of the equilibrium point (pair of price vector and activity vector) are discussed through studying this variational inequality.

General economic equilibrium and variational inequalities

In this paper we apply the theory of variational inequalities to the general economic equilibrium problem. Qualitative properties of existence and uniqueness are established. Then the decomposition algorithm for the computation of the equilibrium point is proposed which decomposes the general economic equilibrium problem into two parts: in the first part, the equilibrium price is determined as a solution of a variational inequality with a compact feasible set; in the second part, the equilibrium activity level is determined as a solution of a set of linear inequalities. Finally, the convergence of the algorithm is established.

The integer linear complementarity problem

In this paper we consider the case of the linear complementarity problem where all or some of the variables are required to take integer values. We discuss several applications to economic equilibrium problems and polymatrix games. When the integer variables are bounded, then the problem can be solved using an equivalent linear integer formulation. For the general problem (unbounded case) the problem can be solved using enumeration of the feasible points of a set of mixed zero-one linear inequalities.

Computational Experience in Solving Equilibrium Models by a Sequence of Linear Complementarity Problems

This paper presents a modeling format and a solution algorithm for partial and general economic equilibrium problems. It reports on computational experience from a series of small to medium sized problems taken from the literature on computation of economic equilibria. The common characteristic of these models is the presence of weak inequalities and complementary slackness, e.g., a linear technology with alternative activities or various institutional constraints on prices. The algorithm computes the equilibrium by solving a sequence of linear complementarity problems. The iterative (outer) part of this algorithm is a Newton process. For the inner part, we use Lemke's almost complementary pivoting algorithm. Theoretical results for the performance of this algorithm are at present available only for the partial equilibrium cases. Our computational experience with both types of models, however, is encouraging. The algorithm solved all nine test problems when initiated at reasonable starting points. Five of these nine problems are solved for several different starting points, indicating a large region over which the algorithm converges. Our results demonstrate that the algorithm is economical in terms of the number of pivots, function evaluations and CPU time.

Sensitivity Analysis for the General Spatial Economic Equilibrium Problem

We consider the general spatial economic equilibrium problem where the supplies and demands at each market for each commodity may depend on the price on this as well as other prices of other commodities at other markets and the transportation cost on each link for each commodity may depend on the flow on this as well as other flows of other commodities on other links in the network. For this model we perform stability and sensitivity analysis. Assuming that the supply, demand, and transportation cost functions are monotone, we first show that the multicommodity equilibrium price pattern depends continuously upon the market supply and demand functions and the multicommodity equilibrium shipment pattern depends continuously upon the transportation cost functions. We then focus on the delicate question of predicting the direction of the change in the price and shipment pattern and the incurred supplies, demands, and transportation costs resulting from changes in the supply, demand, and transportation cost functions and we demonstrate that certain phenomena may prevail. Our analysis depends crucially on the fact that the governing equilibrium conditions can be formulated as a variational inequality.

Stability, sensitivity analysis, and computation of competitive network equilibria : Anna Nagurney, 1983 (Dissertation in the division of Applied Mathematics, Brown University, Providence, RI 02912)

Competitive network equilibrium problems originate in situations where each user of a network acting independently seeks to minimize his personal disutility until, finally, an equilibrium is established characterized by the property that no user has any incentive to change his strategy, given that all other users retain their present strategies. Such competitive equilibrium problems, that arise in economics, operations research, and control theory can be formulated as a variational inequality. Although a plethora of algorithms is available for determining the equilibrium, relatively little effort has been extended towards the computational testing of these algorithms. In addition, the important for the applications questions of stability and sensitivity of the equilibrium have not been addressed yet in any general context. In this thesis we study the stability and sensitivity of the equilibrium for two applications, namely, the traffic equilibrium problem and the spatial economic equilibrium problem. We consider the general asymmetric traffic equilibrium problem with fixed demands. We show that, under certain assumptions, the equilibrium traffic pattern depends continuously upon the assigned travel demands and travel cost functions. We then focus on the delicate question of predicting the direction of the change in the traffic pattern and the incurred travel costs resulting from changes in the travel cost functions and travel demands and attempt to elucidate certain counter-intuitive phenomena such as Braess paradox. Next we consider the general spatial economic equilibrium problem. Under certain conditions we show that the multicommodity equilibrium price pattern depends continuously upon the market supply and demand functions and the multicommodity equilibrium shipment pattern depends continuously upon the transportation cost functions. We then address the important question of predicting the direction of the change in the price and shipment pattern and the incurred supplies, demands, and transportation costs resulting from changes in the supply, demand, and transportation cost functions. Finally, we proceed to test the relative efficiency of the main computational techniques for computing the solution of the variational inequality, the relaxation and projection methods. Our results indicate that the form of certain functions affects the performance of the two basic techniques.

General network equilibrium analysis

Network equilibrium problems, such as electric, hydraulic, traffic and economic equilibrium problems, are very common topics in real world situations. Equilibrium solutions can be characterized by various flow laws which govern the feasible flows on the underlying network, the dependent index quantity which is resulted by feasible flows, and the equilibrium principles which put the network system in a stable state. In this paper, a generalized complementary model is established for the general network equilibrium problems. Due to the nature of conical duality and complementary characterization of equilibrium solutions, under some monotonicity assumptions, the equilibrium problem is equivalent to the extremality conditions of a generalized geometric programming problem. Hence geometric programming methods can be applied to the network equilibrium analysis. Furthermore, the recent studies of Fang and Peterson (1982) in generalized variational inequalities can be adopted to handle more complicated cases. In ...

An algorithm for the generalized transportation problem

This paper describes a new computational technique for solving spatial economic equilibrium problems which are generalizations of the classic transportation problem. This technique makes use of a type of algorithm which has been developed in recent years to compute Kakutani fixed points and solve related problems. Existing algorithms for the generalized transportation problem employ quadratic programming, and therefore require that demand and supply functions be linear. By contrast, the algorithm of this paper can handle demand and supply relationships which are nonlinear or even semi-continuous. It can also handle non-constant transport costs and various other complications. The technique is capable of yielding highly accurate solutions, and appears to be computationally efficient on problems of reasonable size.