Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

Consider the variational inequality VI(C, F) of finding a point x* ∈ C satisfying the property 〈Fx*, x − x*〉≥0, for all x ∈ C, where C is the intersection of finite level sets of convex functions defined on a real Hilbert space H and F : H → H is an L‐Lipschitzian and η‐strongly monotone operator. Relaxed and self‐adaptive iterative algorithms are devised for computing the unique solution of VI(C, F). Since our algorithm avoids calculating the projection PC (calculating PC by computing several sequences of projections onto half‐spaces containing the original domain C) directly and has no need to know any information of the constants L and η, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.