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 → R {\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 f ( x ) = | x | {\displaystyle f(x)=|x|} is non-differentiable when x = 0 {\displaystyle x=0} . However, as seen in the graph on the right (where f ( x ) {\displaystyle f(x)} in blue has non-differentiable kinks similar to the absolute value function), for any x 0 {\displaystyle x_{0}} in the domain of the function one can draw a line which goes through the point ( x 0 , f ( x 0 ) ) {\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.
Definition Rigorously, a subderivative of a convex function f : I → R {\displaystyle f:I\to \mathbb {R} } at a point x 0 {\displaystyle x_{0}} in the open interval I {\displaystyle I} is a real number c {\displaystyle c} such that f ( x ) − f ( x 0 ) ≥ c ( x − x 0 ) {\displaystyle f(x)-f(x_{0})\geq c(x-x_{0})} for all x ∈ I {\displaystyle x\in I} . By the converse of the mean value theorem, the set of subderivatives at x 0 {\displaystyle x_{0}} for a convex function is a nonempty closed interval [ a , b ] {\displaystyle [a,b]} , where a {\displaystyle a} and b {\displaystyle b} are the one-sided limits a = lim x → x 0 − f ( x ) − f ( x 0 ) x − x 0 , {\displaystyle a=\lim _{x\to x_{0}^{-}}{\frac {f(x)-f(x_{0})}{x-x_{0}}},}
b = lim x → x 0 + 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 [ a , b ] {\displaystyle [a,b]} of all subderivatives is called the subdifferential of the function f {\displaystyle f} at x 0 {\displaystyle x_{0}} , denoted by ∂ f ( x 0 ) {\displaystyle \partial f(x_{0})} . If f {\displaystyle f} is convex, then its subdifferential at any point is non-empty. Moreover, if its subdifferential at x 0 {\displaystyle x_{0}} contains exactly one subderivative, then f {\displaystyle f} is differentiable at x 0 {\displaystyle x_{0}} and ∂ f ( x 0 ) = { f ′ ( x 0 ) } {\displaystyle \partial f(x_{0})=\{f'(x_{0})\}} .
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