  We consider the minimum-norm-point (MNP) problem over polyhedra, a
well-studied problem that encompasses linear programming. We present a general
algorithmic framework that combines two fundamental approaches for this
problem: active set methods and first order methods. Our algorithm performs
first order update steps, followed by iterations that aim to `stabilize' the
current iterate with additional projections, i.e., find a locally optimal
solution whilst keeping the current tight inequalities. Such steps have been
previously used in active set methods for the nonnegative least squares (NNLS)
problem.
  We bound on the number of iterations polynomially in the dimension and in the
associated circuit imbalance measure. In particular, the algorithm is strongly
polynomial for network flow instances. Classical NNLS algorithms such as the
Lawson-Hanson algorithm are special instantiations of our framework; as a
consequence, we obtain convergence bounds for these algorithms. Our preliminary
computational experiments show promising practical performance.
