Forward-backward-based descent methods for composite variational inequalities

We consider the monotone composite variational inequality (CVI) where the underlying mapping is formed as the sum of two monotone mappings. We combine the forward-backward and descent direction ideas together, and thus present the unified algorithmic framework of forward-backward-based descent methods for solving the CVI. A new iterate of such a method is generated by a prediction-correction fashion, where the predictor is yielded by the forward-backward method and then the predictor is corrected by a descent step. We derive some implementable forward-backward-based descent algorithms for some concrete cases of the CVI, and verify their numerical efficiency via preliminary numerical experiments.