On the convergence of inertial two-subgradient extragradient method for variational inequality problems

In 2012, Censor et al. proposed the so-called two-subgradient extragradient method (TSEGM) in which both projections have an explicit-formula because the projections are just executed onto half-spaces. However, the convergence of the TSEGM was posted as an open question. In this paper, we answer this open question affirmatively for the case where the closed convex set can be represented as the lower level set of a continuously differentiable convex function. Censor's open question is thus partially answered. More precisely, we combine the inertial technique with the TSEGM and establish the weak convergence of the proposed method under mild conditions.