Clarke generalized gradient

Summary

The Clarke generalized gradient is a set of generalized derivatives of a locally Lipschitz function at a point. It extends both the gradient of a continuously differentiable function and the subdifferential of a convex function. It is always a nonempty, convex and weak-* compact set, and it contains $0$ at every local minimum and every local maximum. Frank Clarke introduced it as the basis of nonsmooth analysis and of optimality conditions for problems without smoothness. The definition gives the precise construction.