Skip to main content


Log in

Inexact implicit methods for monotone general variational inequalities

  • Published:
Mathematical Programming Submit manuscript


Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton–like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems \( Q\big(u^*\big) \in \Omega, \qquad \bigl( v - Q\big(u^*\big) \bigr)^T F\big(u^*\big) \ge 0, \qquad \forall v\in \Omega. \) Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Author information

Authors and Affiliations


Additional information

Received July 31, 1995 / Revised version received January 15, 1999¶ Published online May 28, 1999

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, B. Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 199–217 (1999).

Download citation

  • Issue Date:

  • DOI: