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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 Fenchel's duality theorem ： ウィキペディア英語版 Fenchel's duality theorem In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel. Let ''ƒ'' be a proper convex function on R''n'' and let ''g'' be a proper concave function on R''n''. Then, if regularity conditions are satisfied, :$\backslash min\_x\; (\; f(x)-g(x)\; )\; =\; \backslash max\_p\; (\; g\_\backslash star(p)-f^\backslash star(p)).\backslash ,$ where ''ƒ''  * is the convex conjugate of ''ƒ'' (also referred to as the Fenchel–Legendre transform) and ''g''  * is the concave conjugate of ''g''. That is, :$f^\; \backslash left(\; x^\; \backslash right)\; :=\; \backslash sup\; \backslash left\; \backslash ^n\; \backslash right\backslash \}$ :$g\_\; \backslash left(\; x^\; \backslash right)\; :=\; \backslash inf\; \backslash left\; \backslash ^n\; \backslash right\backslash \}$ ==Mathematical theorem== Let ''X'' and ''Y'' be Banach spaces, $f:\; X\; \backslash to\; \backslash mathbb\; \backslash cup\; \backslash$ and $g:\; Y\; \backslash to\; \backslash mathbb\; \backslash cup\; \backslash$ be convex functions and $A:\; X\; \backslash to\; Y$ be a bounded linear map. Then the Fenchel problems: :$p^$ * = \inf_ \ :$d^$ * = \sup_ \ satisfy weak duality, i.e. $p^$ * \geq d^ *. Note that $f^$ *,g^ * are the convex conjugates of ''f'',''g'' respectively, and $A^$ * is the adjoint operator. The perturbation function for this dual problem is given by $F(x,y)\; =\; f(x)\; +\; g(Ax\; -\; y)$. Suppose that ''f'',''g'', and ''A'' satisfy either # ''f'' and ''g'' are lower semi-continuous and $0\; \backslash in\; \backslash operatorname(\backslash operatornameg\; -\; A\; \backslash operatornamef)$ where $\backslash operatorname$ is the algebraic interior and $\backslash operatornameh$ where ''h'' is some function is the set $\backslash$, or # $A\; \backslash operatornamef\; \backslash cap\; \backslash operatornameg\; \backslash neq\; \backslash emptyset$ where $\backslash operatorname$ are the points where the function is continuous. Then strong duality holds, i.e. $p^$ * = d^ *. If $d^$ * \in \mathbb then supremum is attained. 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「Fenchel's duality 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.