A coordinate descent method for market equilibrium problems with price groups

In the present paper, a model of market equilibrium with price groups in the form of variational inequality for a single-product market of an infinitely divisible product has been considered. Unlike the classical model, in which all market participants are equal and a single equilibrium price is found, it is assumed in this paper that each seller or buyer can split the set of his/her counterparties into non-overlapping groups and assign a certain price function to each group. For this model, the equilibrium conditions have been formulated and proved. The conditions for the existence of a solution to the problem, based on the coercivity property, have been also proposed and justified.
For the model of market equilibrium with price groups, in which the price functions of each seller/buyer for each group depend only on the volume of purchases/sales of this seller/buyer in this group, a method of coordinate descent for finding equilibrium states has been proposed and its convergence has been proved. A series of test calculations have been carried out for problems of different dimension, a comparison of the coordinate descent method with the gradient projection method has been performed, which confirms the efficiency of the proposed method and its promising for further investigation.