Slaters condition and boundary solution

Question

Slater's condition is a sufficient condition for a convex optimization problem to satisfy strong duality. It says that feasible region should have an interior. My question is suppose I have a convex optimization problem which satisfies slater's condition, then can a boundary point be the optima of such a problem, or does slater's also want the optima to be in the interior?

Answer 1

No. It's possible for Slater's condition to be satisfied and the only optimal solution is on the boundary of the feasible region. It's also possible to have Slater's condition be satisfied with the only optimal solution on the interior of the feasible region. Examples are pretty easy to construct.

Answer 2

Slater Constraint qualification has nothing to do with objective function! It is indeed a condition on the constraint set. So for any constraint set you always can construct an objective function whose minimum point lies on boundary or not.