<p>This paper investigates the optimality conditions for mathematical programming problems involving geometric and functional constraints, with objective functions that are Fréchet differentiable and their gradient mappings are locally Lipschitz on an open set. We first establish formulas to compute the asymptotic second-order tangent cone of the constraint sets and decompose the asymptotic second-order tangent cone for the intersection of these sets. We then derive second-order necessary and sufficient optimality conditions for mathematical programming problems. The results are applied to a class of optimal control problems.</p>