Subderivative

In mathematics, subderivatives (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.^[1] Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

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