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

Abstract EN

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.