Inexact implicit methods for monotone general variational inequalities.

*(English)*Zbl 0979.49006Summary: Solving a variational inequality problem is equivalent to finding a solution of a system of non-smooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problems 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

$$Q\left({u}^{*}\right)\in {\Omega},\phantom{\rule{2.em}{0ex}}{(v-Q\left({u}^{*}\right))}^{T}F\left({u}^{*}\right)\ge 0,\phantom{\rule{2.em}{0ex}}\forall v\in {\Omega}\xb7$$

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.

##### MSC:

49J40 | Variational methods including variational inequalities |

90C30 | Nonlinear programming |

90C33 | Complementarity and equilibrium problems; variational inequalities (finite dimensions) |

47J20 | Inequalities involving nonlinear operators |