ABSTRACT
In the last few chapters we saw how fundamental the role of constraint qualification is like the Slater constraint qualification in convex optimization. In Chapter 3 we saw that a relaxation of the Slater constraint qualification to the Abadie constraint qualification leads to an asymptotic version of the KKT conditions for the nonsmooth convex programming problems. Thus it is interesting to ask whether it is possible to develop necessary and sufficient optimality conditions for (CP ) without any constraint qualifications. Recently a lot of work has been done in this respect in the form of sequential optimality conditions. But to the best of our knowledge the first step in this direction was taken by Ben-Tal, Ben-Israel, and Zlobec [7]. They obtained the necessary and sufficient optimality conditions in the smooth scenario in the absence of constraint qualifications. This work was extended to the nonsmooth scenario by Wolkowicz [112]. All these studies involved direction sets, which we will discuss below. So before moving on with the discussion of the results derived by Ben-Tal, Ben-Israel, and Zlobec [7], and Wolkowicz [112], we present the notion of direction sets. Before that we introduce the definition of a blunt cone.
