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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 Quasiconvex function ： ウィキペディア英語版 Quasiconvex function In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form $(-\backslash infty,a)$ is a convex set. Informally, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Quasiconvexity and quasiconcavity extend to functions with multiple arguments the notion of unimodality of functions with a single real argument. ==Definition and properties== A function $f:S\; \backslash to\; \backslash mathbb$ defined on a convex subset ''S'' of a real vector space is quasiconvex if for all $x,\; y\; \backslash in\; S$ and $\backslash lambda\; \backslash in\; ()$ we have : $f(\backslash lambda\; x\; +\; (1\; -\; \backslash lambda)y)\backslash leq\backslash max\backslash big\backslash .$ In words, if ''f'' is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then ''f'' is quasiconvex. Note that the points ''x'' and ''y'', and the point directly between them, can be points on a line or more generally points in ''n''-dimensional space. An alternative way (see introduction) of defining a quasi-convex function $f(x)$ is to require that each sub-levelset $S\_\backslash alpha(f)\; =\; \backslash$ is a convex set. If furthermore : for all $x\; \backslash neq\; y$ and $\backslash lambda\; \backslash in\; (0,1)$, then $f$ is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does. A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function $f$ is quasiconcave if : $f(\backslash lambda\; x\; +\; (1\; -\; \backslash lambda)y)\backslash geq\backslash min\backslash big\backslash .$ and strictly quasiconcave if : $f(\backslash lambda\; x\; +\; (1\; -\; \backslash lambda)y)>\backslash min\backslash big\backslash$ A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets. A function that is both quasiconvex and quasiconcave is quasilinear. A particular case of quasi-concavity, if $S\; \backslash subset\; \backslash mathbb$, is unimodality, in which there is a locally maximal value. 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「Quasiconvex 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.