Final answer:
The subdifferential of a convex function is the set of all subgradients at a point, which are vectors satisfying a specific inequality relating to the function's values. It is a generalization of the derivative, and for differentiable functions, the subdifferential at a point is simply the gradient.
Step-by-step explanation:
The subdifferential of a convex function is a concept from convex analysis, a branch of mathematics. For a given convex function f defined on a real vector space, the subdifferential at a point x is the set of all subgradients at x. A subgradient is any vector g such that for all points y in the domain of f, the inequality f(y) ≥ f(x) + g · (y - x) holds, where ‘·’ denotes the dot product.
Intuitively, the subdifferential can be seen as a generalization of the derivative for functions that may not be differentiable everywhere. Instead of a single slope, a convex function can have a whole set of slopes that satisfy the above inequality, creating the notion of a subdifferential. For differentiable functions, the subdifferential at a point is just the singleton set containing the gradient at that point. However, if the function is not differentiable at a point, the subdifferential will contain all possible slopes of lines that lie below the graph of the function and touch the graph at that point.