Duality and the “ convex ” Karush - Kuhn - Tucker theorem ∗ Erik

Our approach to the Karush-Kuhn-Tucker theorem in [OSC] was entirely based on subdifferential calculus (essentially, it was an outgrowth of the two subdifferential calculus rules contained in the Fenchel-Moreau and Dubovitskii-Milyutin theorems, i.e., Theorems 2.9 and 2.17 of [OSC]). On the other hand, Proposition B.4(v) in [OSC] gives an intimate connection between the subdifferential of a function and the Fenchel conjugate of that function. In the present set of lecture notes this connection forms the central analytical tool by which one can study the connections between an optimization problem and its so-called dual optimization problem (such connections are commonly known as duality relations). We shall first study duality for the convex optimization problem that figured in our Karush-Kuhn-Tucker results. In this simple form such duality is known as Lagrangian duality. Next, in section 2 this is followed by a far-reaching extension of duality to abstract optimization problems, which leads to duality-stability relationships. Then, in section 3 we specialize duality to optimization problems with cone-type constraints, which includes Fenchel duality for semidefinite programming problems.