Schemes for finding minimum-norm solutions of variational inequalities

Consider the variational inequality (VI) of finding a point x∗x∗ such that equation(∗ )x∗∈Fix(T)and〈(I−S)x∗,x−x∗〉≥0,x∈Fix(T) where T,ST,S are nonexpansive self-mappings of a closed convex subset CC of a Hilbert space, and Fix(T)Fix(T) is the set of fixed points of TT. Assume that the solution set ΩΩ of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI (∗); namely, the unique solution x∗x∗ to the quadratic minimization problem: x∗=argminx∈Ω‖x‖2x∗=argminx∈Ω‖x‖2.