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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 Rosenbrock function ： ウィキペディア英語版 Rosenbrock function In mathematical optimization, the Rosenbrock function is a non-convex function used as a performance test problem for optimization algorithms introduced by Howard H. Rosenbrock in 1960. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult. The function is defined by $f(x,\; y)\; =\; (a-x)^2\; +\; b(y-x^2)^2$ It has a global minimum at $(x,\; y)=(a,\; a^2)$, where $f(x,\; y)=0$. Usually $a\; =\; 1$ and $b\; =\; 100$. ==Multidimensional generalisations== Two variants are commonly encountered. One is the sum of $N/2$ uncoupled 2D Rosenbrock problems, : $f(\backslash mathbf)\; =\; f(x\_1,\; x\_2,\; \backslash dots,\; x\_N)\; =\; \backslash sum\_^\; \backslash left(-\; x\_)^2$ + (x_ - 1)^2 \right ). This variant is only defined for even $N$ and has predictably simple solutions. A more involved variant is : $f(\backslash mathbf)\; =\; \backslash sum\_^\; 100\; (x\_\; -\; x\_i^2\; )^2\; +\; (1-x\_i)^2\; \backslash quad\; \backslash mbox\; \backslash quad\; \backslash mathbf\; =\; (\backslash ldots,\; x\_N)\; \backslash in\; \backslash mathbb^N.$ This variant has been shown to have exactly one minimum for $N=3$ (at $(1,\; 1,\; 1)$) and exactly two minima for $4\; \backslash le\; N\; \backslash le\; 7$—the global minimum of all ones and a local minimum near $(x\_1,\; x\_2,\; \backslash dots,\; x\_N)\; =\; (-1,\; 1,\; \backslash dots,\; 1)$. This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of $x$. For small $N$ the polynomials can be determined exactly and Sturm's theorem can be used to determine the number of real roots, while the roots can be bounded in the region of . For larger $N$ this method breaks down due to the size of the coefficients involved. 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「Rosenbrock 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.