An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

In this paper, a new monotonicity, MM-monotonicity, is introduced, and the resolvent operator of an MM-monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI(C,F+G)(C,F+G) and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that FF in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping ∏C(⋅)∏C(⋅) is semismooth, is given for calculating εε-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.