Subgradient

In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to functions which are not differentiable. The subdifferential of a function is set-valued. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

Let f:I→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 nondifferentiable when x=0. However, as seen in the picture on the right, for any x₀ in the domain of the function one can draw a line which goes through the point (x₀, f(x₀)) and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative (because the line is under the graph of f).

Rigorously, a subderivative of a convex function f:I→R at a point x₀ in the open interval I is a real number c such that

for all x in I. One may show that the set of subderivatives at x₀ for a convex function is a nonempty closed interval [a, b], where a and b are the one-sided limits

which are guaranteed to exist and satisfy a ≤ b.

The set [a, b] of all subderivatives is called the subdifferential of the function f at x₀. If f is convex and its subdifferential at ${\displaystyle x_{0}}$ contains exactly one subderivative, then f is differentiable at ${\displaystyle x_{0}}$.

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