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| The '''Symmetric Rank 1''' ('''SR1''') method is a [[quasi-Newton method]] to update the second derivative (Hessian)
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| based on the derivatives (gradients) calculated at two points. It is a generalization to the [[secant method]] for a multidimensional problem.
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| This update maintains the ''symmetry'' of the matrix but does ''not'' guarantee that the update be [[positive definite matrix|''positive definite'']].
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| The sequence of Hessian approximations generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1 method show faster progress towards the true Hessian than do popular alternatives ([[BFGS]] or [[Davidon-Fletcher-Powell formula|DFP]]), in preliminary numerical experiments.<ref name="CGT">{{harvtxt|Conn|Gould|Toint|1991}}</ref> The SR1 method has computational advantages for [[sparsity|sparse]] or [[partial separability|partially separable]] problems.
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| A twice continuously differentiable function <math>x \mapsto f(x)</math> has a [[gradient]] (<math>\nabla f</math>) and [[Hessian matrix]] <math>B</math>: The function <math>f</math> has an expansion as a [[Taylor series]] at <math>x_0</math>, which can be truncated
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| ::<math>f(x_0+\Delta x)=f(x_0)+\nabla f(x_0)^T \Delta x+\frac{1}{2} \Delta x^T {B} \Delta x </math>;
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| its gradient has a Taylor-series approximation also
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| ::<math>\nabla f(x_0+\Delta x)=\nabla f(x_0)+B \Delta x</math>,
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| which is used to update <math>B</math>. The above secant-equation need not have a unique solution <math>B</math>.
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| The SR1 formula computes (via an update of [[Rank (linear algebra)|rank]] 1) the symmetric solution that is closest to the current approximate-value <math>B_k</math>:
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| ::<math>B_{k+1}=B_{k}+\frac {(y_k-B_k \Delta x_k) (y_k-B_k \Delta x_k)^T}{(y_k-B_k \Delta x_k)^T \Delta x_k}</math>,
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| where
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| ::<math>y_k=\nabla f(x_k+\Delta x_k)-\nabla f(x_k)</math>.
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| The corresponding update to the approximate inverse-Hessian <math>H_k=B_k^{-1}</math> is
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| ::<math>H_{k+1}=H_{k}+\frac {(\Delta x_k-H_k y_k)(\Delta x_k-H_k y_k)^T}{(\Delta x_k-H_k y_k)^T y_k}</math>.
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| The SR1 formula has been rediscovered a number of times. A drawback is that the denominator can vanish. Some authors have suggested that the update be applied only if
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| ::<math>|\Delta x_k^T (y_k-B_k \Delta x_k)|\geq r \|\Delta x_k\|\cdot \|y_k-B_k \Delta x_k\| </math>, | |
| where <math>r\in(0,1)</math> is a small number, e.g. <math>10^{-8}</math>.<ref>{{harvtxt|Nocedal|Wright|1999}}</ref>
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| ==See also==
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| * [[Quasi-Newton method]]
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| * [[Newton's method in optimization]]
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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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| ==Notes==
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| <references/>
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| ==References==
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| * Byrd, Richard H. (1996) Analysis of a Symmetric Rank-One Trust Region Method. ''SIAM Journal on Optimization'' 6(4)
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| * {{cite journal|last1=Conn|first1=A. R.|last2=
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| Gould|first2=N. I. M.|last3=Toint|first3=Ph. L.|title=Convergence of quasi-Newton matrices generated by the symmetric rank one update|journal=Mathematical Programming|date=March 1991|publisher=Springer Berlin/ Heidelberg|
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| issn=0025-5610|pages=177–195|volume=50|number=1|doi=10.1007/BF01594934|ref=harv|id=[ftp://ftp.numerical.rl.ac.uk/pub/nimg/pubs/ConnGoulToin91_mp.pdf PDF file at Nick Gould's website]|mr=}}
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| * Khalfan, H. Fayez (1993) A Theoretical and Experimental Study of the Symmetric Rank-One Update. ''SIAM Journal on Optimization'' 3(1)
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| * Nocedal, Jorge & Wright, Stephen J. (1999). ''Numerical Optimization''. Springer-Verlag. ISBN 0-387-98793-2.
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| {{Optimization algorithms|unconstrained}}
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| [[Category:Optimization algorithms and methods]]
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