subdifferentiable

Subdifferentiability is a concept in mathematics, specifically in the field of convex analysis, that generalizes the notion of differentiability to non-differentiable functions. It is particularly useful in optimization and variational analysis, where functions may not be smooth.

A function f: R^n → R is said to be subdifferentiable at a point x if there exists a subgradient of f at x. A subgradient of f at x is a vector g such that for all y in R^n, the following inequality holds:

f(y) ≥ f(x) + <g, y - x>

where <g, y - x> denotes the dot product of g and (y - x). The set of all subgradients at x is called the subdifferential of f at x and is denoted by ∂f(x).

If f is convex, then the subdifferential ∂f(x) is non-empty for all x in the domain of f. Moreover, if f is differentiable at x, then ∂f(x) contains only the gradient of f at x. However, for non-differentiable convex functions, the subdifferential can contain multiple vectors, reflecting the fact that the function may have multiple "directions" of increase at a given point.

Subdifferentiability plays a crucial role in the study of convex optimization problems, where the goal is to minimize a convex function. The subdifferential can be used to characterize the optimality conditions for such problems, and it provides a way to generalize the concept of the gradient to non-differentiable functions.