Optimality conditions and a barrier method in optimization with convex geometric constraint

In this note we address a new look to some questions raised by Lasserre in his works (Optim. Lett. 4:1–5, 2010, Optim. Lett. 5:549–556, 2011), concerning the preservation of the conclusions of some results in smooth convex optimization with inequalities constraints to the case where the feasible set is convex, but has no convex representation. The main results we show concern, on one hand, some relations between the hypotheses imposed by Lasserre and the Mangasarian–Fromowitz condition, and, on the other hand, a barrier method based only on the geometric representation of the feasible set.