In this chapter we study optimization problems involving convex functions that are not necessarily di erentiable. Naturally, several new tools are needed in order to compensate for the lack of di erentiability. On one hand, we need to study convex sets which were brie y de ned and studied in Section 2.1. A new object to replace the di erential is introduced and studied. With all these tools in hand, we will be able to derive a generalized Karush-Kuhn-Tucker theorem in the case of convex nonsmooth optimization.