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

Abstract: In this paper, a new monotonicity, -monotonicity, is introduced, and the resolvent operator of an -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 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 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 is semismooth, is given for calculating -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable. [Copyright &y& Elsevier]