Regularity Conditions via Quasi-Relative Interior in Convex Programming

We give some new regularity conditions for Fenchel duality in separated locally convex vector spaces, written in terms of the notion of quasi interior and quasi-relative interior, respectively. We provide also an example of a convex optimization problem for which the classical generalized interior-point conditions given so far in the literature cannot be applied, while the one given by us is applicable. By using a technique developed by Magnanti, we derive some duality results for the optimization problem with cone constraints and its Lagrange dual problem, and we show that a duality result recently given in the literature for this pair of problems has self-contradictory assumptions.