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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 Convex optimization ： ウィキペディア英語版 Convex optimization Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum. Given a real vector space $X$ together with a convex, real-valued function :$f:\backslash mathcal\backslash to\; \backslash mathbb$ defined on a convex subset $\backslash mathcal$ of $X$, the problem is to find any point $x^\backslash ast$ in $\backslash mathcal$ for which the number $f(x)$ is smallest, i.e., a point $x^\backslash ast$ such that :$f(x^\backslash ast)\; \backslash le\; f(x)$ for all $x\; \backslash in\; \backslash mathcal$. The convexity of $f$ makes the powerful tools of convex analysis applicable. In finite-dimensional normed spaces, the Hahn–Banach theorem and the existence of subgradients lead to a particularly satisfying theory of necessary and sufficient conditions for optimality, a duality theory generalizing that for linear programming, and effective computational methods. Convex minimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics (optimal design), and finance. With recent improvements in computing and in optimization theory, convex minimization is nearly as straightforward as linear programming. Many optimization problems can be reformulated as convex minimization problems. For example, the problem of ''maximizing'' a ''concave'' function ''f'' can be re-formulated equivalently as a problem of ''minimizing'' the function -''f'', which is ''convex''. ==Convex optimization problem== An ''optimization problem'' (also referred to as a ''mathematical programming problem'' or ''minimization problem'') of finding some $x^\backslash ast\; \backslash in\; \backslash mathcal$ such that :$f(x^\backslash ast)\; =\; \backslash min\; \backslash ,$ where $\backslash mathcal\; \backslash subset\; \backslash mathbb^n$ is the ''feasible set'' and $f(x):\backslash mathbb^n\; \backslash rightarrow\; \backslash mathbb$ is the ''objective'', is called ''convex'' if $\backslash mathcal$ is a closed convex set and $f(x)$ is convex on $\backslash mathbb^n$. Alternatively, an optimization problem of the form : $\backslash begin$ &\operatorname& & f(x) \\ &\operatorname & &g_i(x) \leq 0, \quad i = 1,\dots,m \end is called convex if the functions $f,\; g\_1\; \backslash ldots\; g\_m\; :\; \backslash mathbb^n\; \backslash rightarrow\; \backslash mathbb$ are convex.〔Boyd/Vandenberghe, p. 7〕 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「Convex optimization」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.