翻訳と辞書 Words near each other ・ Logarithmic form ・ Logarithmic growth ・ Logarithmic integral function ・ Logarithmic mean ・ Logarithmic mean temperature difference ・ Logarithmic norm ・ Logarithmic number system ・ Logarithmic pair ・ Logarithmic resistor ladder ・ Logarithmic scale ・ Logarithmic Schrödinger equation ・ Logarithmic spiral ・ Logarithmic spiral beaches ・ Logarithmic timeline ・ Logarithmic units ・ Logarithmically concave function ・ Logarithmically concave measure ・ Logarithmically concave sequence ・ Logarithmically convex function ・ Logarithmically convex set ・ Logarji ・ Logarovci ・ Logarska Dolina ・ Logaršče ・ Logashkino ・ Logatec ・ Logatec Karst Field ・ Logau-Las ・ Logaščica ・ Logba Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Logarithmically concave function ： ウィキペディア英語版 Logarithmically concave function In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality : $$ f(\theta x + (1 - \theta) y) \geq f(x)^ f(y)^ for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, : $$ \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y) for all and . Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality : $$ f(\theta x + (1 - \theta) y) \leq f(x)^ f(y)^ for all and . ==Properties== * A positive log-concave function is also quasi-concave. * Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function  =  which is log-concave since  =  is a concave function of . But is not concave since the second derivative is positive for || > 1: ::$f\text{'}\text{'}(x)=e^\}\; (x^2-1)\; \backslash nleq\; 0$ * A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all satisfying , ::$f(x)\backslash nabla^2f(x)\; \backslash preceq\; \backslash nabla\; f(x)\backslash nabla\; f(x)^T$,〔Stephen Boyd and Lieven Vandenberghe, (Convex Optimization ) (PDF) p.105〕 :i.e. ::$f(x)\backslash nabla^2f(x)\; -\; \backslash nabla\; f(x)\backslash nabla\; f(x)^T$ is :negative semi-definite. For functions of one variable, this condition simplifies to ::$f(x)f\text{'}\text{'}(x)\; \backslash leq\; (f\text{'}(x))^2$ 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「Logarithmically concave function」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.