In this paper a new algorithm to minimize convex functions is developed. This algorithm is based on the concept of codifferential. Since the computation of whole codifferential is not always possible we propose an algorithm for computation of descent directions using only a few elements from the codifferential. The convergence of the proposed minimization algorithm is proved and results of numerical experiments using a set of test problems with nonsmooth convex objective function are reported. We also compare the proposed algorithm with three different versions of bundle methods. Tins comparison shows that the proposed method is more robust than bundle methods.