Spatial Price Equilibrium Problem Research Articles

A stochastic fractional differential variational inequality with Lévy jump and its application

This paper investigates a stochastic fractional differential variational inequality with Lévy jump (SFDVI with Lévy jump), which comprises a stochastic fractional differential equation with Lévy jump and a stochastic variational inequality. By employing the successive approximation method as well as the projection technique, we establish unique existence of the solution for the SFDVI with Lévy jump under some mild conditions. Moreover, the main results are applied to obtain the unique existence of the solution for the spatial price equilibrium problem in stochastic environments.

Optimal commodity price stabilization as a multi-period spatial equilibrium problem: A supernetwork approach with public buffer stocks

This paper addresses the seasonal agricultural commodity price stabilization problem both with and without price bands using a spatial and temporal supernetwork framework. The problem is formulated as an extension of the classic spatial price equilibrium problem and solved with the gradient projection algorithm. Results from the model support the use of interseasonal storage mechanisms and confirm that the use of price bands is less effective as it lowers total society surplus while providing inaccurate signaling and therefore dampening market perceptions of product scarcity or excess. An illustrative numerical example provides a comparison between policy alternatives.

User Equilibrium Exchange Allocation Algorithm Based on Super Network

The theory of super network is an effective method to various traffic networks with means of multiple decision-making. It provides us with a favorable pricing decision tool for it combines a practical transport network with the space pricing decision. Spatial price equilibrium problem has always been the important research direction of the Transport Economics and regional transportation planning. As to how to combine the two, this paper presents the user equilibrium exchange allocation algorithm based on super network, which successfully keep the classical spatial price equilibrium problems (SPE) into a super-network analysis framework. Through super-network analysis, we can add two virtual nodes in the network, which correspond to the virtual supply node and the super-super-demand virtual node, analysis the user equivalence with the SPE equilibrium and find the concrete steps of users exchange allocation algorithm based on super-network equilibrium. Finally, we carried out experiments to verify. The experiments show that: through the user equilibrium exchange SPE allocation algorithm based on super-network, we can get the steady-state equilibrium solution, which demonstrate that the algorithm is reasonable.

A PROGRESSIVE EQUILIBRATION ALGORITHM FOR GENERAL MARKET EQUILIBRIUM PROBLEMS UNDER CONSTRAINTS

This paper considers the general Spatial Price Equilibrium (SPE) problem under constraints of supply-guarantee at demand markets and, in the meantime, protecting the scarce resources at supply markets. We first formulate the equilibrium problem as a variational inequality (VI) problem with partially unknown mapping. Then, we propose a progressive equilibration algorithm for solving the problem under consideration. The global convergence of the proposed method is proved under suitable assumptions. Some preliminary numerical results demonstrate the reliability of the proposed algorithm.

Special issue on applications of variational inequality problems

I am very pleased to see the completion of this special issue of Optimization and Engineering: Applications of Variational Inequality Problems. Over the past several decades, variational inequalities have attracted a great deal of attention from both researchers and practitioners. It is well known that variational inequalities cover diverse disciplines in partial differential equations, optimal control, optimization, mathematical programming, mechanics, finance, operations research, engineering (dynamic rigidbody model, nonlinear obstacle problems), economics equilibrium models, and science. This special issue focus on applications of variational inequality problems. It includes 10 papers selected after a peer revision. The summaries of the 10 papers in this issue are listed below: The first paper, by Laura Scrimali, discusses time-dependent spatial price equilibrium problem. The author presents two different perspectives. First a useroptimization time-dependent model is studied. In particular, the equilibrium conditions are stated and the equivalence between the problem under consideration and an evolutionary variational inequality is proved. Afterwards, introducing control policies, the problem becomes a time-dependent spatial price equilibrium control problem from the policy-maker’s point of view and it is formulated as an evolutionary

Dynamics and equilibria of ecological predator–prey networks as nature’s supply chains

In this paper, we develop a dynamic network model of ecological food webs and prove that the set of stationary points of the projected dynamical system coincides with the set of solutions of a variational inequality governing the equilibrium of predator–prey networks. We also establish the equivalence between the ecological models and supply chain network equilibrium models and highlight the connections to spatial price equilibrium problems. We propose an algorithmic scheme, provide convergence results, and apply it to a food web drawn from a fisheries application. This paper is a contribution to the interdisciplinary supply chain network literature.

Dynamics and Equilibria of Ecological Predator-Prey Networks as Nature’s Supply Chains

Ecological food webs in the form of predator-prey networks are nature’s supply chains. Here, we develop a dynamic network model of ecological food webs and prove that the set of stationary points of the projected dynamical system coincides with the set of solutions of a variational inequality governing the equilibrium of predator-prey networks. We also establish the relationship between the ecological models and supply chain network equilibrium models and highlight the connections to spatial price equilibrium problems. We propose an algorithmic scheme and provide convergence results. Finally, we apply the framework to a food web drawn from a fisheries application.This paper is a contribution to the literature on the interfaces among economics, transportation and logistics, and biology/ecology through network models and methodologies.

An inverse variational inequality approach to the evolutionary spatial price equilibrium problem

It is well known that the time-dependent spatial price equilibrium problem can be transformed into and studied as an evolutionary variational inequality. However, in some situations, control policies may be imposed to the end of regulating the amounts of production and consumption. As a consequence, the problem becomes a time-dependent spatial price equilibrium control problem and is formulated as an evolutionary inverse variational inequality. The existence of solutions is then investigated and a numerical example is also provided.

Dynamic spatial price equilibrium problem: regularity results and computational procedure

AbstractIn this paper we consider the dynamic spatial price equilibrium problem and the associated evolutionary variational inequality. In particular the regularity of equilibrium solutions is studied and a method for solving the dynamic spatial price equilibrium problem is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Sensitivity analysis for time dependent spatial price equilibrium problem

We present some sensitivity results for the spatial price equilibrium problem in the case of quantity formulation model and in presence of excess supply and excess demand. The equilibrium conditions that describe the above model are expressed in terms of a time dependent variational inequality. The variational inequality formulation plays a fundamental role in order to achieve the sensitivity results.

Time Dependent Spatial Price EquilibriumProblems

We consider a time dependent economic market in order to show the existence of the dynamical market equilibrium. Moreover, the Generalized Lagrange Multipliers and the Lagrangian theories are studied and, as an interesting consequence, we obtain the Lagrangianvariables. This theory plays an extraordinaryrole in the estimation of the dynamicalmarket equilibrium.In our opinion this study makes these problems more general and realistic than classical static spatial price equilibrium problems.

Bicriterion Weight Varying Spatial Price Networks

We consider a spatial price equilibrium problem in which the consumers take their decisions according to the transportation cost and transportation time necessary for obtaining a given commodity. In particular, each consumer market can give a different weight to each component of a generalized cost, and we suppose that this weight can depend on time. Thus, we are faced with a time-dependent equilibrium problem which we cast within the framework of variational inequalities. We give existence results and, by using the example of a linear operator, we propose also a discretization procedure for equilibrium problems which can be modeled by the same type of variational inequality.

Time—Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Model

This paper concerns with the study of the time–dependent variational inequality associated to the spatial price equilibrium model related to the quantity formulation. In particular existence the -orems of the solution to the associated variational inequality and a stability analysis of the equilibrium pattern is reported.

New Alternating Direction Method for a Class of Nonlinear Variational Inequality Problems

The alternating direction method is an attractive method for a class of variational inequality problems if the subproblems can be solved efficiently. However, solving the subproblems exactly is expensive even when the subproblem is strongly monotone or linear. To overcome this disadvantage, this paper develops a new alternating direction method for cocoercive nonlinear variational inequality problems. To illustrate the performance of this approach, we implement it for traffic assignment problems with fixed demand and for large-scale spatial price equilibrium problems.

Strategic gaming analysis for electric power systems: an MPEC approach

Transmission constraints and market concentration may prevent power markets from being fully competitive, allowing firms to exercise market power and raise prices above marginal cost. We present a strategic gaming model for analyzing such markets; it represents an oligopolistic market economy consisting of several dominant firms in an electric power network. Each generating firm submits bids to an ISO, choosing its bids to maximize profits subject to anticipated reactions by rival firms. The single-firm model is formulated as a mathematical program with equilibrium constraints (MPEC) with a parameter-dependent spatial price equilibrium problem as the inner problem. Power flows and pricing strategies are constrained by the ISO's linearized DC optimal power flow (OFF) model. A penalty interior point algorithm is used to compute a local optimal solution of the MPEC. Numerical examples based on a 30 bus network are presented, including multi-firm Nash equilibria in which each player solves an MPEC of the single-firm type.

A class of projection and contraction methods for asymmetric linear variational inequalities and their relations to Fukushima's descent method

For solving asymmetric linear variational inequalities, we present a class of projection and contraction methods under the general G-norm. The search direction of our methods is just a convex combination of two descent directions of Fukushima's merit function. However, we use the direction to reduce the distance function ( 1 2 )∥u − u∗∥ 2 G , where μ∗ is a solution point of the problem. Finally, we report some numerical results for spatial price equilibrium problems by using the presented methods.

On the stability of an adjustment process for spatial price equilibrium modeled as a projected dynamical system

Abstract This paper presents a stability study for the continuous-time adjustment process proposed for the perfectly competitive spatial price equilibrium problem modeled as a projected dynamical system by Nagurney, Takayama, and Zhang (1995). Two distinct approaches are adopted here for establishing the stability of the spatial price equilibrium under such an adjustment process. One approach utilizes the monotonicity condition on the supply prices, demand prices, and transaction costs to develop various stability properties. The other utilizes the regularity concept, adapted from Zhang and Nagurney (1995), which enables us to explore the stability of the adjustment process — a nonstandard dynamical system, via its minimal face flow — a standard dynamical system.

On truncated‐newton methods for solving the spatial price equilibrium problem

AbstractA truncated‐Newton method is applied to the spatial price equilibrium problem and the method is compared with other specialized techniques that exploit the inherent network structure of the problem. Variants of the linear conjugate gradient algorithm for approximately solving the Newton equations are discussed. Numerical results for the truncated‐Newton method are compared with those of a block‐SOR method. Tests indicate that the two approaches are comparable when transportation costs are linear. A truncated‐Newton algorithm is far superior if the transportation costs are nonlinear.

Mathematical modelling of spatial price equilibrium for multicommodity consumer flows of large markets using variational inequalities

This paper presents an additional application of patient/customer flow models within a large geographical domain to that which were presented in previous articles. Some additional calibration results are also provided for an origin-constrained model using real data from a hospital database for the state of Massachusetts. The equivalence of an equity objective function model that was presented in earlier papers is made to the spatial price equilibrium (SPE) problem. The corresponding mathematical formulations for the latter and an optimization algorithm for its solution are presented for patient/client/customer flows. Finally a Cournot-Nash oligopolistic model in which a few hospitals are competing in spatially separated markets is also formulated, as well as its corresponding solution using the mathematical principle of variational inequalities (VI). Conclusions and future directions of the research are discussed.

Massively parallel computation of spatial price equilibrium problems as dynamical systems

Abstract In this paper we introduce a dynamical system for the formulation and computation of spatial price equilibrium problems in quantity variables. The set of stationary points of the system corresponds to the set of solutions of the variational inequality problem governing the problem. We propose the Euler-type method for the computation of the equilibrium pattern and provide convergence results. We then demonstrate that the algorithm can be implemented on a massively parallel architecture and illustrate its performance on the Thinking Machine's CM-2 architecture. This research represents the first implementation of a massively parallel approach for the computation of either dynamical systems or variational inequality problems arising in economics.