On second-order local minimum conditions for smooth optimization problems with abstract constraints

For the problem of minimizing a twice Freshet differentiable function f: X → R over an abstract set Q ⊂ X we supplement known second order necessary local minimum conditions with a new one which should be satisfied by nonzero second order asymptotic tangent vectors to Q. We also prove that in finite-dimensional settings the collection consisting of strengthened (known and new) necessary conditions is sufficient for local strict minimum in the problem under consideration.