Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets

In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second order sufficient conditions. We show that the second order regularity condition always holds in the case of semidefinite programming.