ABSTRACT
In this chapter we are going to look into a completely different approach to develop optimality conditions in convex programming. These optimality conditions, called sequential optimality conditions, can hold without any qualification and thus both from a theoretical as well as practical point of view this is of great interest. To the best of our knowledge, this approach was initiated by Thibault [108]; Jeyakumar, Rubinov, Glover, and Ishizuka [70]; and Jeyakumar, Lee, and Dinh [68]. Unlike the approach of direction sets in Chapter 6, in the sequential approach one needs calculus rules for subdifferentials and εsubdifferentials, namely the Sum Rule and the Chain Rule. As the name itself suggests, the sequential optimality conditions are established as a sequence of subdifferentials at neighborhood points as in the work of Thibault [108] or sequence of ε-subdifferentials at the exact point as in the study of Jeyakumar and collaborators [68, 70]. Thibault [108] used the approach of sequential subdifferential calculus rules while Jeyakumar and collaborators [68, 70] used the approach of epigraphs of conjugate functions to study the sequential optimality conditions extensively. In both these approaches, the convex programming problem involved cone constraints and abstract constraints. But keeping in sync with the convex programming problem (CP ) studied in this book, we consider the feasible set C involving convex inequalities. The reader must have realized the central role of the Slater constraint qualification in the study of optimality and duality in optimization. However, as we have seen, the Slater constraint qualification can fail even for very simple problems. The failure of the Slater constraint qualification was overcome by the development of the so-called closed cone constraint qualification. It is a geometric qualification that uses the Fenchel conjugate of the constraint function. We will study this qualification condition in detail.
