Limiting subdifferential

Summary

The limiting subdifferential, also called the Mordukhovich or basic subdifferential, is a set of generalized gradients of a function at a point. It is built in two steps. One first takes the regular (Fréchet) subgradients at nearby points, then all limits of these as the points approach the given point. For a convex function it equals the subdifferential of convex analysis, and for a locally Lipschitz function its convex hull is the Clarke generalized gradient. It is usually smaller than the Clarke generalized gradient and has a good calculus, which makes it a main tool of variational analysis. The definition gives the precise construction.