Necessary conditions for the optimality of a pair ((y) over bar, (u) over bar) with respect to a locally Lipschitz cost functional L(y, u), subject to Ay + F(y) = Cu + B(u), are given in terms of generalized gradients, Here A and C are densely defined, closed, linear operators on some Banach spaces, while F and B are (Frechet) differentiable maps, which are suitably related to A and C. Various examples and potential applications to nonlinear programming models and nonlinear optimal control of partial differential equations are also discussed.