Subderivative

Summary

Take a convex function and a point of its graph. A subderivative there is the slope of a line through that point that stays below the whole graph. At every interior point a convex function has at least one subderivative, even where it has no derivative. The set of all subderivatives at a point is the subdifferential, and it replaces the derivative in the optimality conditions of nonsmooth convex optimization. The definition gives the precise condition.