subdifferentiability

Subdifferentiability is a concept in mathematics, particularly in the field of convex analysis, that generalizes the notion of differentiability to non-differentiable functions. It is particularly useful in the study of convex functions, where the subdifferential provides a way to extend the concept of the gradient to functions that may not be differentiable at every point.

The subdifferential of a convex function f at a point x, denoted by ∂f(x), is the set of all subgradients of f at x. A subgradient of f at x is a vector v such that f(y) ≥ f(x) + v^T(y - x) for all y in the domain of f. This inequality is known as the subgradient inequality.

If the subdifferential ∂f(x) is non-empty at a point x, then f is said to be subdifferentiable at x. If f is subdifferentiable at every point in its domain, then f is said to be subdifferentiable.

Subdifferentiability is a weaker condition than differentiability. For example, a convex function is always subdifferentiable, but it may not be differentiable at some points. The subdifferential can be used to study the behavior of convex functions, such as their optimality conditions and convergence properties in optimization algorithms.