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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 : 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 : : 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 contains exactly one subderivative, then ''f'' is differentiable at .〔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
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