Inexact decomposition methods for solving variational inequalities with linear equality constraint

Abstract

In this paper, we propose two inexact decomposition methods for solving variational inequalities(VI) with linear equality constraint, which improve the decomposition method proposed by Gabay (in Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian methods: applications to the solution of boundary-valued problems, pp. 299–331, North-Holland, Amsterdam, 1983), Gabay and Mercier (Comput. Math. Appl. 2(1):17–40, 1976) in the following two senses: in each iteration, both methods allow the involved strongly monotone sub-VI to be solved approximately; the temporal iterate generated by the sub-VI is utilized to generate descent direction, and the new iterate is generated along the descent direction. Under mild conditions, the global convergence of the inexact methods is proved. Some numerical experiments are carried out to validate the efficiency and practicality of the proposed methods.