Penalty and path-following methods have been used for solving finitedimensional quadratic programmes. The intention here is to apply such techniques to an infinite-dimensional problem, namely a one-sided obstacle problem, and to develop a method for solving the problem in an infinite-dimensional setting. The numerical methods developed in the infinite-dimensional context, so the convergence rate of the discretisations are (in some sense) independent to the size of the finite-dimensional approximation. These methods are shown to be convergent in appropriate Banach spaces by means of a monotonicity result for the iterates of the associated Newton method. This montonicity carries over to finite-dimensional discretisations for a large class of methods. The overall numerical method developed is based on an exterior penalty fund;ion, and some num."erical results have been obtained.