On Second Order Sufficient Conditions for a Minimum

The paper offers second order necessary and sufficient conditions for a sufficiently general class of optimization problems in Banach spaces. The reader can easily see that this is a ``no gap'' pair of conditions, although sufficient conditions (as usual) hold under some additional assumptions, mainly aimed to guarantee that tangent sets to the non-functional constraint set approximate the latter sufficiently well. In the last section we apply the results to a mathematical programming problem that contains, along with functional equality and inequality constraints, also a non-functional set constraint.