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  "body": "\n## Summary\n\nThe Clarke generalized gradient is a set of generalized derivatives of a locally Lipschitz function at a point.\nIt extends both the gradient of a continuously differentiable function and the [[subdifferential|9C5WXYYQEK]] of a convex function.\nIt is always a nonempty, convex and weak-* compact set, and it contains $0$ at every local minimum and every local maximum.\nFrank Clarke introduced it as the basis of nonsmooth analysis and of optimality conditions for problems without smoothness.\nThe [[definition|NCWYTR6PH0]] gives the precise construction.\n\n## Explanations\n\n### L1 human\n\nA function like $\\abs{x}$ has a corner, so at $0$ it has no derivative.\nNear the corner its slopes are $-1$ on the left and $1$ on the right.\nThe Clarke generalized gradient collects all slopes that appear as limits near the point and fills in the gaps between them.\nFor $\\abs{x}$ at $0$ this gives the interval $[-1, 1]$.\nIt gives the same interval for $-\\abs{x}$, which has a peak at $0$, so it does not distinguish minima from maxima.\n",
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