Consider the variational inequality (VI) of finding a point x∗ such that (∗ )x∗∈Fix(T)and〈(I−S)x∗,x−x∗〉≥0,x∈Fix(T) where T,S are nonexpansive self-mappings of a closed convex subset C of a Hilbert space, and Fix(T) is the set of fixed points of T. Assume that the solution set Ω 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∗ to the quadratic minimization problem: x∗=argminx∈Ω‖x‖2.