Parametric Disjunctive Programming: One-Sided Differentiability of the Value Function

We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory.For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.