subderivative

right|thumb|A convex function (blue) and "subtangent lines" at x_0 (red). In mathematics, the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

Let f:I \to \mathbb{R} 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 f(x)=|x| is non-differentiable when x=0. However, as seen in the graph on the right (where f(x) in blue has non-differentiable kinks similar to the absolute value function), for any x_0 in the domain of the function one can draw a line which goes through the point (x_0,f(x_0)) and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative.