subdifferentialen
Subdifferentialen, typically called the subdifferential in English, is a concept in convex analysis that generalizes the gradient to nonsmooth functions. It collects all linear lower bounds of a function at a given point and provides a way to describe slopes when a function is not differentiable.
For a proper lower semicontinuous convex function f: R^n → R ∪ {∞}, the subdifferential at x is defined
∂f(x) = { g ∈ R^n | f(y) ≥ f(x) + g^T (y − x) for all y }.
Geometrically, ∂f(x) is the set of slopes of all supporting hyperplanes to the graph of f at
Key properties include that ∂f(x) is a convex and closed set. If f is differentiable at x,
In optimization, a point x minimizes f if and only if 0 ∈ ∂f(x). This condition underpins many
Generalizations exist for nonconvex or nonsmooth settings. Fréchet and limiting (Mordukhovich) subdifferentials extend the concept beyond