Blind signal separation consists in processing a set of observed mixed signals in order to separate them into a set of components without any a priori knowledge about the mixing process. This paper deals with the blind separation of nonnegative signals. We show that, for such signals, the problem can be expressed as the identification of relevant extreme directions of a data defined polyhedral cone. Direction relevance is determined by means of a new criterion which integrates both sparseness and linear independence. In order to optimize this criterion with a low complexity, a suboptimal but efficient algorithm based on linear programming is proposed. After a rigorous soundness proof, the steps of the proposed algorithm are detailed, its convergence is analyzed and its performance is evaluated via experiments involving two-dimensional signals.