---
tag: 20967BXZKA
kind: concept
name: Clarke generalized gradient
msc: [49J52, 26B25]
sources:
  - url: https://en.wikipedia.org/wiki/Clarke_generalized_derivative
same_as:
  wikidata: Q116858622
  wikipedia: Clarke generalized derivative
---

## Summary

The Clarke generalized gradient is a set of generalized derivatives of a locally Lipschitz function at a point.
It extends both the gradient of a continuously differentiable function and the [[subdifferential|9C5WXYYQEK]] of a convex function.
It is always a nonempty, convex and weak-* compact set, and it contains $0$ at every local minimum and every local maximum.
Frank Clarke introduced it as the basis of nonsmooth analysis and of optimality conditions for problems without smoothness.
The [[definition|NCWYTR6PH0]] gives the precise construction.

## Explanations

### L1 human

A function like $\abs{x}$ has a corner, so at $0$ it has no derivative.
Near the corner its slopes are $-1$ on the left and $1$ on the right.
The Clarke generalized gradient collects all slopes that appear as limits near the point and fills in the gaps between them.
For $\abs{x}$ at $0$ this gives the interval $[-1, 1]$.
It gives the same interval for $-\abs{x}$, which has a peak at $0$, so it does not distinguish minima from maxima.