The following nonsmooth constraint minimization problem is considered: Minimize f(x) subject to \(x\in X_ 0\), -g(x)\(\in S\), \(h(x)=0\), where \(X_ 0\) is an open subset of \(R^ n\), \(f: X_ 0\to R\), \(g: K_ 0\to R^ m\), \(h: X_ 0\to R^ r\) are locally Lipschitz functions, not necessarily Fréchet-differentiable, S is an arbitrary closed convex cone in \(R^ m\), int \(S\neq \emptyset\). Necessary Lagrangian conditions for this problem are obtained via a sequence of smooth local approximations to the problem. The approach is applied to finite dimensional problems with convex cones, semiinfinite programming problems, continuous programming problems, and optimal control. A duality theorem is obtained, in which the usual convexity hypotheses are weakened to a generalized invex condition using generalized gradients.