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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Danskin's theorem ： ウィキペディア英語版 Danskin's theorem In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form :$f(x)\; =\; \backslash max\_\; \backslash phi(x,z).$ The theorem has applications in optimization, where it sometimes is used to solve minimax problems. The original theorem by J. M. Danskin, given in his 1967, monograph "The Theory of Max-Min and its Applications to Weapons Allocation Problems," Springer, NY, provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function. When adapted to the case of a convex function, this formula yields the following theorem given in somewhat more general form as Proposition A.22 in the 1971 Ph.D. Thesis by D. P. Bertsekas, "Control of Uncertain Systems with a Set-Membership Description of the Uncertainty". A proof of the following version can be found in the 1999 book "Nonlinear Programming" by Bertsekas (Section B.5). == Statement == The theorem applies to the following situation. Suppose $\backslash phi(x,z)$ is a continuous function of two arguments, :$\backslash phi:\; ^n\; \backslash times\; Z\; \backslash rightarrow$ where $Z\; \backslash subset\; ^m$ is a compact set. Further assume that $\backslash phi(x,z)$ is convex in $x$ for every $z\; \backslash in\; Z$. Under these conditions, Danskin's theorem provides conclusions regarding the differentiability of the function :$f(x)\; =\; \backslash max\_\; \backslash phi(x,z).$ To state these results, we define the set of maximizing points $Z\_0(x)$ as :$Z\_0(x)\; =\; \backslash left\backslash )\; =\; \backslash max\_\; \backslash phi(x,z)\backslash right\backslash \}.$ Danskin's theorem then provides the following results. ;Convexity : $f(x)$ is convex. ;Directional derivatives : The directional derivative of $f(x)$ in the direction $y$, denoted $D\_y\backslash \; f(x)$, is given by ::$D\_y\; f(x)\; =\; \backslash max\_\; \backslash phi\text{'}(x,z;y),$ : where $\backslash phi\text{'}(x,z;y)$ is the directional derivative of the function $\backslash phi(\backslash cdot,z)$ at $x$ in the direction $y$. ;Derivative : $f(x)$ is differentiable at $x$ if $Z\_0(x)$ consists of a single element $\backslash overline$. In this case, the derivative of $f(x)$ (or the gradient of $f(x)$ if $x$ is a vector) is given by :: $\backslash frac\; =\; \backslash frac.$ ;Subdifferential :If $\backslash phi(x,z)$ is differentiable with respect to $x$ for all $z\; \backslash in\; Z$, and if $\backslash partial\; \backslash phi/\backslash partial\; x$ is continuous with respect to $z$ for all $x$, then the subdifferential of $f(x)$ is given by :: $\backslash partial\; f(x)\; =\; \backslash mathrm\; \backslash left\backslash \; :\; z\; \backslash in\; Z\_0(x)\; \backslash right\backslash \}$ : where $\backslash mathrm$ indicates the convex hull operation. ;Extension The 1971 Ph.D. Thesis by Bertsekas 〔(Ph.D. Thesis by D. P. Bertsekas )〕 (Proposition A.22) proves a more general result, which does not require that $\backslash phi(\backslash cdot,z)$ is differentiable. Instead it assumes that $\backslash phi(\backslash cdot,z)$ is an extended real-valued closed proper convex function for each $z$ in the compact set $Z$, that $int(dom(f))$, the interior of the effective domain of $f$, is nonempty, and that $\backslash phi$ is continuous on the set $int(dom(f))\backslash times\; Z$. Then for all $x$ in $int(dom(f))$, the subdifferential of $f$ at $x$ is given by :: $\backslash partial\; f(x)\; =\; \backslash mathrm\; \backslash left\backslash$ where $\backslash partial\; \backslash phi(x,z)$ is the subdifferential of $\backslash phi(\backslash cdot,z)$ at $x$ for any $z$ in $Z$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Danskin's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.