  In the present paper, we focus on the vector optimization problems with
inequality constraints, where objective functions and constrained functions are
Fr\'echet differentiable, and whose gradient mapping is locally Lipschitz on an
open set. By using the second-order symmetric subdifferential and the
second-order tangent set, we propose two types of second-order regularity
conditions in the sense of Abadie. Then we establish some strong second-order
Karush--Kuhn--Tucker necessary optimality conditions for Geoffrion properly
efficient solutions of the considered problem. Examples are given to illustrate
the obtained results.
