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  "body": "\n## Definition\n\nLet $f \\colon \\R^n \\to (-\\infty, +\\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]], and let $\\bar x \\in \\R^n$ with $f(\\bar x)$ finite.\n\n1. A vector $v \\in \\R^n$ is a *regular subgradient* of $f$ at a point $x$ with $f(x)$ finite if\n$$\\liminf_{y \\to x, \\ y \\ne x} \\frac{f(y) - f(x) - \\ip{v}{y - x}}{\\abs{y - x}} \\ge 0,$$\nwhere $\\liminf$ is the [[limit inferior|033RZ32T4Q]].\nThe set of these vectors is the *regular subdifferential* $\\hat\\partial f(x)$.\n2. The *limiting subdifferential* $\\partial f(\\bar x)$ is the set of vectors $v \\in \\R^n$ for which there are [[sequences|0G9EKM0E15]] $x_k \\to \\bar x$ and $v_k \\to v$ with $f(x_k) \\to f(\\bar x)$ and $v_k \\in \\hat\\partial f(x_k)$ for all $k$.\n\n## Remarks\n\nTaking $x_k = \\bar x$ and $v_k = v$ shows that $\\hat\\partial f(\\bar x) \\subseteq \\partial f(\\bar x)$.\nIf $f$ is differentiable near $\\bar x$ with continuous gradient, then $\\partial f(\\bar x) = \\set{\\nabla f(\\bar x)}$.\nIf $f$ is convex, both sets equal the [[subdifferential|9C5WXYYQEK]] of convex analysis.\nIf $\\bar x$ is a local minimum of $f$, then $0 \\in \\hat\\partial f(\\bar x) \\subseteq \\partial f(\\bar x)$.\nOther names are the Mordukhovich subdifferential and the basic subdifferential.\n\nExample: $f(x) = -\\abs{x}$ on $\\R$ and $\\bar x = 0$.\nFor $x \u003e 0$ the function is differentiable with derivative $-1$, so $\\hat\\partial f(x) = \\set{-1}$; for $x \u003c 0$, $\\hat\\partial f(x) = \\set{1}$.\nHence $\\set{-1, 1} \\subseteq \\partial f(0)$, and in fact $\\partial f(0) = \\set{-1, 1}$.\n\nNon-example: the vector $v = 0$ for the same function.\nIt is not a regular subgradient at $0$, since $(f(y) - f(0) - 0)/\\abs{y} = -1$ for all $y \\ne 0$.\nIt is not a limit of regular subgradients at nearby points either, since these are all $\\pm 1$.\nSo $0 \\notin \\partial f(0)$, although $0$ lies in the Clarke generalized gradient $[-1, 1]$.\n",
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