A new alternating projection-based prediction–correction method for structured variational inequalities. (4th July 2019) - Tu, K.; Zhang, H. B.; Xia, F. Q.
Abstract : Recently, some proximal-based alternating direction methods and alternating projection-based prediction–correction methods were proposed to solve the structured variational inequalities in Euclidean space R n . We note that the proximal-based alternating direction methods need to solve its subproblems exactly. However, the subproblems of the proximal-based alternating direction methods are too difficult to be solved exactly in many practical applications. We also note that the existing alternating projection based prediction–correction methods just can cope with the case that the underlying mappings are Lipschitz continuous. However, it could be difficult to verify their Lipschitz continuity condition, provided that the available information is only the mapping values. In this paper, we present a new alternating projection-based prediction–correction method for solving the structured variational inequalities, where the underlying mappings are continuous. In each iteration, we first employ a new Armijo linesearch to derive the predictors, and then update the next iterate via some minor computations. Under some mild assumptions, we establish the global convergence theorem of the proposed method. Preliminary numerical results are also reported to illustrate the effectiveness of the proposed method.