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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 Wolfe duality ： ウィキペディア英語版 Wolfe duality In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle. == Mathematical formulation == For a minimization problem with inequality constraints, : $\backslash begin$ &\underset & &g_i(x) \leq 0, \quad i = 1,\dots,m \end the Lagrangian dual problem is : $\backslash begin$ &\underset^m u_j g_j(x)\right) \\ &\operatorname & &u_i \geq 0, \quad i = 1,\dots,m \end where the objective function is the Lagrange dual function. Provided that the functions $f$ and $g\_1,\; \backslash ldots,\; g\_m$ are continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem : $\backslash begin$ &\underset^m u_j g_j(x) \\ &\operatorname & & \nabla f(x) + \sum_^m u_j \nabla g_j(x) = 0 \\ &&&u_i \geq 0, \quad i = 1,\dots,m \end is called the Wolfe dual problem. This problem employs the KKT conditions as a constraint. This problem may be difficult to deal with computationally, because the objective function is not concave in the joint variables $(u,x)$. Also, the equality constraint $\backslash nabla\; f(x)\; +\; \backslash sum\_^m\; u\_j\; \backslash nabla\; g\_j(x)$ is nonlinear in general, so the Wolfe dual problem is typically a nonconvex optimization problem. In any case, weak duality holds. 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「Wolfe duality」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.