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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 Nonlinear conjugate gradient method ： ウィキペディア英語版 Nonlinear conjugate gradient method In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function $\backslash displaystyle\; f(x)$: :: $\backslash displaystyle\; f(x)=\backslash |Ax-b\backslash |^2$ The minimum of $f$ is obtained when the gradient is 0: :: $\backslash nabla\_x\; f=2\; A^\backslash top(Ax-b)=0$. Whereas linear conjugate gradient seeks a solution to the linear equation $\backslash displaystyle\; A^\backslash top\; Ax=A^\backslash top\; b$, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient $\backslash nabla\_x\; f$ alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum. Given a function $\backslash displaystyle\; f(x)$ of $N$ variables to minimize, its gradient $\backslash nabla\_x\; f$ indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction: :: $\backslash Delta\; x\_0=-\backslash nabla\_x\; f\; (x\_0)$ with an adjustable step length $\backslash displaystyle\; \backslash alpha$ and performs a line search in this direction until it reaches the minimum of $\backslash displaystyle\; f$: :: $\backslash displaystyle\; \backslash alpha\_0:=\; \backslash arg\; \backslash min\_\backslash alpha\; f(x\_0+\backslash alpha\; \backslash Delta\; x\_0)$, :: $\backslash displaystyle\; x\_1=x\_0+\backslash alpha\_0\; \backslash Delta\; x\_0$ After this first iteration in the steepest direction $\backslash displaystyle\; \backslash Delta\; x\_0$, the following steps constitute one iteration of moving along a subsequent conjugate direction $\backslash displaystyle\; s\_n$, where $\backslash displaystyle\; s\_0=\backslash Delta\; x\_0$: # Calculate the steepest direction: $\backslash Delta\; x\_n=-\backslash nabla\_x\; f\; (x\_n)$, # Compute $\backslash displaystyle\; \backslash beta\_n$ according to one of the formulas below, # Update the conjugate direction: $\backslash displaystyle\; s\_n=\backslash Delta\; x\_n+\backslash beta\_n\; s\_$ # Perform a line search: optimize $\backslash displaystyle\; \backslash alpha\_n=\backslash arg\; \backslash min\_\; f(x\_n+\backslash alpha\; s\_n)$, # Update the position: $\backslash displaystyle\; x\_=x\_+\backslash alpha\_\; s\_$, With a pure quadratic function the minimum is reached within N iterations (excepting roundoff error), but a non-quadratic function will make slower progress. Subsequent search directions lose conjugacy requiring the search direction to be reset to the steepest descent direction at least every N iterations, or sooner if progress stops. However, resetting every iteration turns the method into steepest descent. The algorithm stops when it finds the minimum, determined when no progress is made after a direction reset (i.e. in the steepest descent direction), or when some tolerance criterion is reached. Within a linear approximation, the parameters $\backslash displaystyle\; \backslash alpha$ and $\backslash displaystyle\; \backslash beta$ are the same as in the linear conjugate gradient method but have been obtained with line searches. The conjugate gradient method can follow narrow (ill-conditioned) valleys where the steepest descent method slows down and follows a criss-cross pattern. Four of the best known formulas for $\backslash displaystyle\; \backslash beta\_n$ are named after their developers and are given by the following formulas: * Fletcher–Reeves: :: $\backslash beta\_^\; =\; \backslash frac$ } * Polak–Ribière: :: $\backslash beta\_^\; =\; \backslash frac$ } * Hestenes-Stiefel: :: $\backslash beta\_n^\; =\; -\backslash frac$ )} * Dai–Yuan: :: $\backslash beta\_^\; =\; -\backslash frac$ )} . These formulas are equivalent for a quadratic function, but for nonlinear optimization the preferred formula is a matter of heuristics or taste. A popular choice is $\backslash displaystyle\; \backslash beta=\backslash max\backslash$ which provides a direction reset automatically. Newton based methods - Newton-Raphson Algorithm, Quasi-Newton methods (e.g., BFGS method) - tend to converge in fewer iterations, although each iteration typically requires more computation than a conjugate gradient iteration as Newton-like methods require computing the Hessian (matrix of second derivatives) in addition to the gradient. Quasi-Newton methods also require more memory to operate (see also the limited memory L-BFGS method). ==See also== * Nelder–Mead method * conjugate gradient method 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「Nonlinear conjugate gradient method」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.