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 as

∂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 x.

Key properties include that ∂f(x) is a convex and closed set. If f is differentiable at x, then ∂f(x) = {∇f(x)}. If x lies in the interior of the domain of f, the subdifferential is nonempty; outside the domain it may be empty. Examples: for f(x) = |x| on R, ∂f(0) = [-1, 1] and ∂f(a) = {sign(a)} for a ≠ 0; for f(x) = x^2, ∂f(x) = {2x}.

In optimization, a point x minimizes f if and only if 0 ∈ ∂f(x). This condition underpins many nonsmooth optimization algorithms, such as subgradient methods.

Generalizations exist for nonconvex or nonsmooth settings. Fréchet and limiting (Mordukhovich) subdifferentials extend the concept beyond convex functions, while the Clarke subdifferential applies to Lipschitz functions, capturing a broader notion of generalized gradients.