This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in \cA} (B^a x - b^a)=0$ where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and $\cA$ is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem.