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| The '''Davidon–Fletcher–Powell formula''' (or '''DFP'''; named after [[William C. Davidon]], [[Roger Fletcher (mathematician)|Roger Fletcher]], and [[Michael J. D. Powell]]) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition (see below). It was the first [[quasi-Newton method]] which generalize the [[secant method]] to a multidimensional problem. This update maintains the symmetry and positive definiteness of the [[Hessian matrix]].
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| Given a function <math>f(x)</math>, its [[gradient]] (<math>\nabla f</math>), and [[positive definite matrix|positive definite]] [[Hessian matrix]] <math>B</math>, the [[Taylor series]] is:
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| :<math>f(x_k+s_k)=f(x_k)+\nabla f(x_k)^T s_k+\frac{1}{2} s^T_k {B} s_k, </math> | |
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| and the [[Taylor series]] of the gradient itself (secant equation):
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| :<math>\nabla f(x_k+s_k)=\nabla f(x_k)+B s_k,</math> | |
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| is used to update <math>B</math>.
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| The DFP formula finds a solution that is symmetric, positive definite and closest to the current approximate value of <math>B_k</math>:
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| :<math>B_{k+1}=
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| (I-\gamma_k y_k s_k^T) B_k (I-\gamma_k s_k y_k^T)+\gamma_k y_k y_k^T,</math>
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| where
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| :<math>y_k=\nabla f(x_k+s_k)-\nabla f(x_k),</math>
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| :<math>\gamma_k =\frac{1}{y_k^T s_k}.</math>
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| and <math>B_k</math> is a symmetric and [[positive definite matrix]].
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| The corresponding update to the inverse Hessian approximation <math>H_k=B_k^{-1}</math> is given by:
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| :<math>H_{k+1}=H_{k}-\frac{H_k y_k y_k^T H_k}{y_k^T H_k y_k}+\frac{s_k s_k^T}{y_k^{T} s_k}.</math> | |
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| <math>B</math> is assumed to be positive definite, and
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| the vectors <math>s_k^T</math> and <math>y</math> must satisfy the curvature condition:
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| : <math>s_k^T y_k=s_k^T B s_k>0. \, </math>
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| The DFP formula is quite effective, but it was soon superseded by the [[BFGS method|BFGS formula]], which is its dual (interchanging the roles of y and s).
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| ==See also==
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| * [[Newton's method]]
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| * [[Newton's method in optimization]]
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| * [[Quasi-Newton method]]
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| * [[BFGS method|Broyden–Fletcher–Goldfarb–Shanno (BFGS) method]]
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| * [[L-BFGS|L-BFGS method]]
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| * [[SR1 formula]]
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| * [[Nelder–Mead method]]
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| ==References==
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| * {{Citation |doi=10.1137/0801001 |first1=W. C.|last1= Davidon|title=Variable metric method for minimization|journal= SIAM Journal on Optimization |volume=1|pages=1–17 |year=1991}}
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| * {{Citation | last1=Fletcher | first1=Roger | title=Practical methods of optimization | publisher=John Wiley & Sons | location=New York | edition=2nd | isbn=978-0-471-91547-8 | year=1987}}.
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| * {{Citation|author=Nocedal, Jorge & Wright, Stephen J. |year=1999|title=Numerical Optimization|publisher= Springer-Verlag| isbn= 0-387-98793-2}}
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| {{Optimization algorithms}}
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| {{DEFAULTSORT:Davidon-Fletcher-Powell formula}}
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| [[Category:Optimization algorithms and methods]]
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Friends call him Royal. Interviewing is what I do for a residing but I plan on altering it. Delaware has usually been my living place and will by no means move. Camping is something that I've carried out for many years.
Also visit my blog :: http://ktva-Online.com/index.php?mod=users&action=view&id=12078