A weak constraint qualification for conic programs and a problem on duality gap

We discuss a weak constraint qualification for conic linear programs and its applications for a few classes of cones. This constraint qualification is used to give a solution to a problem proposed by Shapiro and Z\v{a}linescu and show that if a closed convex cone is such that all its non-trivial faces are polyhedral and all the non-trivial exposed faces of its dual are polyhedral, then the duality gap is zero as long as the primal and dual problems are feasible. Moreover, the common optimal value must be attained at least at one of the sides. We also show an example of a cone that meets the requirements our theorem but is such that previously known results cannot be used to prove its good duality properties.