On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions. © 2006 Springer-Verlag

Application of Market Models to Network Equilibrium Problems

We present a general two-side market model with divisible commodities and price functions of participants. A general existence result on unbounded sets is obtained from its variational inequality re-formulation. We describe an extension of the network flow equilibrium problem with elastic demands and a new equilibrium type model for resource allocation problems in wireless communication networks, which appear to be particular cases of the general market model. This enables us to obtain new existence results for these models as some adjustments of that for the market model. Under certain additional conditions the general market model can be reduced to a decomposable optimization problem where the goal function is the sum of two functions and one of them is convex separable, whereas the feasible set is the corresponding Cartesian product. We discuss some versions of the partial linearization method, which can be applied to these network equilibrium problems.Comment: 18 pages, 3 table

Application of market models to network equilibrium problems

© Springer International Publishing AG, part of Springer Nature 2018. We present a general two-side market model with divisible commodities and price functions of participants. A general existence result on unbounded sets is obtained from its variational inequality reformulation. We describe an extension of the network flow equilibrium problem with elastic demands and a new equilibrium type model for resource allocation problems in wireless communication networks, which appear to be particular cases of the general market model. This enables us to obtain new existence results for these models as some adjustments of that for the market model. Under certain additional conditions, the general market model can be reduced to a decomposable optimization problem where the goal function is the sum of two functions and one of them is convex separable, whereas the feasible set is the corresponding Cartesian product. We discuss some versions of the partial linearization method, which can be applied to these network equilibrium problems

Variational Inequality Type Formulations of General Market Equilibrium Problems with Local Information

We suggest a new approach to creation of general market equilibrium models involving economic agents with local and partial knowledge about the system and under different restrictions. The market equilibrium problem is then formulated as a quasi-variational inequality that enables us to establish existence results for the model in different settings. We also describe dynamic processes, which fall into information exchange schemes of the proposed market model. In particular, we propose an iterative solution method for quasi-variational inequalities, which is based on evaluations of the proper market information only in a neighborhood of the current market state without knowledge of the whole feasible set and prove its convergence.Comment: 22 page

Descent and penalization techniques for equilibrium problems with nonlinear constraints

This paper deals with equilibrium problems with nonlinear constraints. Exploiting a gap function recently introduced, which rely on a polyhedral approximation of the feasible region, we propose two descent methods. They are both based on the minimization of a suitable exact penalty function, but they use different rules for updating the penalization parameter and they rely on different types of line search. The convergence of both algorithms is proved under standard assumptions

Application of Market Equilibrium Models to Optimal Resource Allocation in Telecommunication Networks

Abstract: We consider several problems of optimal resource allocation in telecommunication networks and show that they can be formulated as market equilibrium models. This approach enables us to create simple and efficient solution methods. Next, we consider such a resource allocation problem for a provider of a wireless communication network divided into zones (clusters). The network manager aims to distribute some homogeneous resource (bandwidth) among users of several zones in order to maximize the total network profit, which takes into account payments from users and implementation costs. As a result, we obtain a convex optimization problem involving capacity and balance constraints. By using the dual Lagrangian method with respect to the capacity constraint, we reduce the initial problem to a suitable one-dimensional problem, so that calculation of its cost function value leads to independent solution of zonal problems, treated as two-side market equilibrium models with one trader. We show that solution of each zonal problem can be found exactly by a simple arrangement type algorithm even in the case where the trader price is not fixed. Besides, we suggest ways to adjust the basic problem to the case of moving nodes. Some results of computational experiments confirm the applicability of the new method

Existence and solution methods for equilibria

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them

Merit functions: a bridge between optimization and equilibria

In the last decades, many problems involving equilibria, arising from engineering, physics and economics, have been formulated as variational mathematical models. In turn, these models can be reformulated as optimization problems through merit functions. This paper aims at reviewing the literature about merit functions for variational inequalities, quasi-variational inequalities and abstract equilibrium problems. Smoothness and convexity properties of merit functions and solution methods based on them will be presented

On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions. © 2006 Springer-Verlag

On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions. © 2006 Springer-Verlag