On subdifferential calculus ∗

The main purpose of these lectures is to familiarize the student with the basic ingredients of convex analysis, especially its subdifferential calculus. This is done while moving to a clearly discernible end-goal, the Karush-Kuhn-Tucker theorem, which is one of the main results of nonlinear programming. Of course, in the present lectures we have to limit ourselves most of the time to the Karush-Kuhn-Tucker theorem for convex nonlinear programming. While this is on the one hand restrictive, it is somewhat compensated for by extra structure that the Karush-Kuhn-Tucker theory gains in the presence of convexity. The material is presented in the following way. It is assumed that several – but perhaps not all – students have already been exposed to some standard material on convex sets. This material has been collected in the appendix; it will be referred to during the lectures whenever the need arises. Sometimes further references will be given; as a rule these concern results that can be found in the textbooks [1] or [2]. The less standard part of the material, notably subdifferential calculus, is treated in the main part of the text.