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 ${\displaystyle f:I\to \mathbb {R} }$ 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.

Rigorously, a subderivative of a convex function ${\displaystyle f:I\to \mathbb {R} }$ at a point ${\displaystyle x_{0}}$ in the open interval ${\displaystyle I}$ is a real number ${\displaystyle c}$ such that $${\displaystyle f(x)-f(x_{0})\geq c(x-x_{0})}$$for all ${\displaystyle x\in I}$. By the converse of the mean value theorem, the set of subderivatives at ${\displaystyle x_{0}}$ for a convex function is a nonempty closed interval ${\displaystyle [a,b]}$, where ${\displaystyle a}$ and ${\displaystyle b}$ are the one-sided limits$${\displaystyle a=\lim _{x\to x_{0}^{-}}{\frac {f(x)-f(x_{0})}{x-x_{0}}},}$$$${\displaystyle b=\lim _{x\to x_{0}^{+}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}.}$$The interval ${\displaystyle [a,b]}$ of all subderivatives is called the subdifferential of the function ${\displaystyle f}$ at ${\displaystyle x_{0}}$, denoted by ${\displaystyle \partial f(x_{0})}$. If ${\displaystyle f}$ is convex, then its subdifferential at any point is non-empty. Moreover, if its subdifferential at ${\displaystyle x_{0}}$ contains exactly one subderivative, then ${\displaystyle f}$ is differentiable at ${\displaystyle x_{0}}$ and ${\displaystyle \partial f(x_{0})=\{f'(x_{0})\}}$.

Consider the function ${\displaystyle f(x)=|x|}$ which is convex. Then, the subdifferential at the origin is the interval ${\displaystyle [-1,1]}$. The subdifferential at any point ${\displaystyle x_{0}<0}$ is the singleton set ${\displaystyle \{-1\}}$, while the subdifferential at any point ${\displaystyle x_{0}>0}$ is the singleton set ${\displaystyle \{1\}}$. This is similar to the sign function, but is not single-valued at ${\displaystyle 0}$, instead including all possible subderivatives.

The concepts of subderivative and subdifferential can be generalized to functions of several variables. If ${\displaystyle f:U\to \mathbb {R} }$ is a real-valued convex function defined on a convex open set in the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, a vector ${\displaystyle v}$ in that space is called a subgradient at ${\displaystyle x_{0}\in U}$ if for any ${\displaystyle x\in U}$ one has that

where the dot denotes the dot product. The set of all subgradients at ${\displaystyle x_{0}}$ is called the subdifferential at ${\displaystyle x_{0}}$ and is denoted ${\displaystyle \partial f(x_{0})}$. The subdifferential is always a nonempty convex compact set.

These concepts generalize further to convex functions ${\displaystyle f:U\to \mathbb {R} }$ on a convex set in a locally convex space ${\displaystyle V}$. A functional ${\displaystyle v^{*}}$ in the dual space ${\displaystyle V^{*}}$ is called a subgradient at ${\displaystyle x_{0}}$ in ${\displaystyle U}$ if for all ${\displaystyle x\in U}$,

The set of all subgradients at ${\displaystyle x_{0}}$ is called the subdifferential at ${\displaystyle x_{0}}$ and is again denoted ${\displaystyle \partial f(x_{0})}$. 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 ${\displaystyle f}$ is continuous, the subdifferential is nonempty.