Let be a
real-valued convex function defined on an
open interval of the real line. Such a function need not be differentiable at all points: For example, the
absolute value function is non-differentiable when . However, as seen in the graph on the right (where in blue has non-differentiable kinks similar to the absolute value function), for any in the domain of the function one can draw a line which goes through the point and which is everywhere either touching or below the graph of f. The
slope of such a line is called a subderivative.
Rigorously, a subderivative of a convex function at a point in the open interval is a real number such that
The set of all subderivatives is called the subdifferential of the function at , denoted by . If is convex, then its subdifferential at any point is non-empty. Moreover, if its subdifferential at contains exactly one subderivative, then and is differentiable at .
Consider the function which is convex. Then, the subdifferential at the origin is the interval . The subdifferential at any point is the
singleton set, while the subdifferential at any point is the singleton set . This is similar to the
sign function, but is not single-valued at , instead including all possible subderivatives.
A convex function is differentiable at if and only if the subdifferential is a singleton set, which is .
A point is a
global minimum of a convex function if and only if zero is contained in the subdifferential. For instance, in the figure above, one may draw a horizontal "subtangent line" to the graph of at . This last property is a generalization of the fact that the derivative of a function differentiable at a local minimum is zero.
If and are convex functions with subdifferentials and with being the interior point of one of the functions, then the subdifferential of is (where the addition operator denotes the
Minkowski sum). This reads as "the subdifferential of a sum is the sum of the subdifferentials."
The concepts of subderivative and subdifferential can be generalized to functions of several variables. If is a real-valued convex function defined on a
convexopen set in the
Euclidean space, a vector in that space is called a subgradient at if for any one has that
where the dot denotes the
The set of all subgradients at is called the subdifferential at x0 and is denoted . The subdifferential is always a nonempty convex
The set of all subgradients at is called the subdifferential at and is again denoted . The subdifferential is always a convex
closed set. It can be an empty set; consider for example an
unbounded operator, which is convex, but has no subgradient. If is continuous, the subdifferential is nonempty.
The subdifferential on convex functions was introduced by
Jean Jacques Moreau and
R. Tyrrell Rockafellar in the early 1960s. The generalized subdifferential for nonconvex functions was introduced by F.H. Clarke and R.T. Rockafellar in the early 1980s.