Novel projected gradient methods for solving pseudomontone variational inequality problems with applications to optimal control problems

In this paper, we consider the projected gradient method for solving strongly pseudo-monotone, Lipschitz continuous variational inequalities in Hilbert spaces. Under strong pseudomonotonicity and Lipschitz continuity hypotheses, we prove the sequence generated by the proposed algorithms converges to the unique solution of VIs with an $ R $-linear rate without the prior knowledge of the Lipschitz constant of the variational inequality mapping. Moreover, the self-adaptive step size in the algorithms is bounded from below by a positive number this has not been studied before for the projected gradient method. Some numerical result applications in optimal control problems are provided to demonstrate the advantages and efficiency of the proposed algorithms over some existing results.