On Dynamical Equilibrium Problems and Variational Inequalities

Abstract

We consider a time-dependent economic market in order to show the existence of time-dependent market equilibrium (which we call dynamic equilibrium). The model we are concerned with is the spatial price equilibrium model in the presence of excesses of supplies and of demands.This kind of network problem is directly incorporated into the Variational Inequality model, which provides not only the existence, but also the computation, the stability and the sensitivity of the equilibrium patterns.The study of the time-dependent model seems to be important because it allows us to follow the evolution in time of prices and of commodities.

Weak and Strong Convergence of Subgradient Extragradient Methods for Pseudomonotone Equilibrium Problems

“…(b) We consider the bifunction f : C × C → ℜ which is generalized from the Cournot-Nash model [14,25] as (12) f (x, y) = P x + Qy + q, y − x , where q ∈ ℜ m and P, Q are two m × m matrices. Since T n is a half-space, the optimization problem in Step 2 of Algorithm 3.1 is always a convex quadratic problem (only with one linear inequality constraint).…”

The extragradient-Armijo method for pseudomonotone equilibrium problems and strict pseudocontractions

“…, where N C (x) is out normal cone atx on C and ∂g(·) denotes the subdifferential of g (see [20]). Now we are in a position to describe the extragradient-Armijo algorithm for finding…”

Strong Convergence of an Extended Extragradient Method for Equilibrium Problems and Fixed Point Problems

“…Lemma 2.3 (see [5]). Let C be a nonempty closed convex subset of a real Hilbert space H and g : C → R be convex and subdifferentiable on C. Then x * is a solution to the following convex problem…”

“…These problems appear frequently in many practical problems arising, for instance, physics, engineering, game theory, transportation, economics and network, and become an attractive field for many researchers both theory and applications (see [1,2,3,4,5,18,21]). If f (x, y) = ⟨F (x), y − x⟩ for every x, y ∈ C, where F is a mapping from C to H, then the problem EP (f, C) becomes the following variational inequality: Find x * ∈ C such that For solving V I(F, C) in the Euclidean space R n under the assumption that a subset C ⊆ R n is nonempty closed convex, F is monotone, L-Lipschitz continuous and Sol(F, C) ̸ = ∅, Korpelevich in [9] introduced the following extragradient method:…”