  We consider an optimization problem subject to an abstract constraint and
finitely many nonlinear constraints. Using the recently introduced concept of
$n$-polyhedricity, we are able to provide second-order optimality conditions
under weak regularity assumptions. In particular, we prove necessary optimality
conditions of first and second order under the constraint qualification of
Robinson, Zowe and Kurcyusz. Similarly, sufficient optimality conditions are
stated. The gap between both conditions is as small as possible.
