In order to maximize a piecewise affine concave function one can basically use the simplex method or an interior point method. As an alternative, we propose the face simplex method. The vertex to vertex scheme of the simplex method is replaced by a more general face to face scheme in the face simplex method. To improve the current iterate, in the face simplex method, one computes the steepest ascent on the current face of the objective function graph and then an exact linesearch determines next iterate. This new procedure can be used in the cutting plane framework as a substitute of the simplex method. As a preliminary numerical test, this new version of the cutting plane method is compared with three other methods : subgradient, Kelley cutting plane and ACCPM.