In this paper we present a review of the development of the discrete maximum principle. In the presentation, the emphasis is on a geometrical interpretation. The crucial assumptions in the theory developed are pointed out, and the attempts to overcome the limitation in the resulting theorems ore exposed. Following the review, we present a new approach to optimization of the multi-stage optimization problems called the ’ upper boundary approach ’. The classical methods of solving this problem are shown to fit smoothly into the new approach. Moreover, using this approach a number of new results have been developed, among these a new generalized version of the discrete maximum principle. The new version does not require the assumption of directional convexity.