Some Goldsteinʹs type methods for co-coercive variant variational inequalities

The classical Goldsteinʹs method has been well studied in the context of variational inequalities (VIs). In particular, it has been shown in the literature that the Goldsteinʹs method works well for co-coercive VIs where the underlying mapping is co-coercive. In this paper, we show that the Goldsteinʹs method can be extended to solve co-coercive variant variational inequalities (VVIs). We first show that when the Goldsteinʹs method is applied to solve VVIs, the iterative scheme can be improved by identifying a refined step-size if the involved co-coercive modulus is known. By doing so, the allowable range of the involved scaling parameter ensuring convergence is enlarged compared to that in the context of VVIs with Lipschitz and strongly monotone operators. Then, we show that for such a VVI whose co-coercive modulus is unknown, the Goldsteinʹs method is still convergent provided that an easily-implementable Armijoʹs type strategy of adjusting the scaling parameter self-adaptively is employed. Some numerical results are reported to verify that the proposed Goldsteinʹs type methods are efficient for solving VVIs.