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  "body": "\n## Summary\n\nTake a [[convex function|07Q7CNM5WF]] and a point of its graph.\nA subderivative there is the slope of a line through that point that stays below the whole graph.\nAt every interior point a convex function has at least one subderivative, even where it has no [[derivative|0V1EJCT8G6]].\nThe set of all subderivatives at a point is the subdifferential, and it replaces the derivative in the optimality conditions of nonsmooth convex optimization.\nThe [[definition|A289X08MR1]] gives the precise condition.\n\n## Explanations\n\n### L1 human\n\nAt a smooth point of a convex graph there is one tangent line, and it lies below the graph.\nAt a corner there are many lines that touch the graph there and stay below it.\nTheir slopes are the subderivatives.\nFor $\\abs{x}$ at $0$, every slope between $-1$ and $1$ works, so the subdifferential is $[-1, 1]$.\nA point is a minimum of a convex function exactly when $0$ is one of the subderivatives there.\n",
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