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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 Subderivative ： ウィキペディア英語版 Subderivative In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to functions which are not differentiable. The subdifferential of a function is set-valued. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let ''f'':''I''→R be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function ''f''(''x'')=|''x''| is nondifferentiable when ''x''=0. However, as seen in the picture on the right, for any ''x''0 in the domain of the function one can draw a line which goes through the point (''x''0, ''f''(''x''0)) and which is everywhere either touching or below the graph of ''f''. The slope of such a line is called a ''subderivative'' (because the line is under the graph of ''f''). ==Definition== Rigorously, a ''subderivative'' of a function ''f'':''I''→R at a point ''x''0 in the open interval ''I'' is a real number ''c'' such that :$f(x)-f(x\_0)\backslash ge\; c(x-x\_0)$ for all ''x'' in ''I''. One may show that the set of subderivatives at ''x''0 for a convex function is a nonempty closed interval (''b'' ), where ''a'' and ''b'' are the one-sided limits :$a=\backslash lim\_\backslash frac$ :$b=\backslash lim\_\backslash frac$ which are guaranteed to exist and satisfy ''a'' ≤ ''b''. The set (''b'' ) of all subderivatives is called the subdifferential of the function ''f'' at ''x''0. If ''f'' is convex and its subdifferential at $x\_0$ contains exactly one subderivative, then ''f'' is differentiable at $x\_0$.〔R. T. Rockafellar ''Convex analysis'' 1970. Theorem 25.1, p.242〕 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「Subderivative」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.