The Symmetric Rank 1 (SR1) method is a quasi-Newton method to update the second derivative (Hessian)
based on the derivatives (gradients) calculated at two points. It is a generalization to the secant method for a multidimensional problem.
This update maintains the symmetry of the matrix but does not guarantee that the update be positive definite.
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 DFP), in preliminary numerical experiments.[1][2] The SR1 method has computational advantages for sparse or partially separable problems.[3]
A twice continuously differentiable function
has a gradient (
) and Hessian matrix
: The function
has an expansion as a Taylor series at
, which can be truncated
;
its gradient has a Taylor-series approximation also
,
which is used to update
. The above secant-equation need not have a unique solution
.
The SR1 formula computes (via an update of rank 1) the symmetric solution that is closest[further explanation needed] to the current approximate-value
:
,
where
.
The corresponding update to the approximate inverse-Hessian
is
.
One might wonder why positive-definiteness is not preserved — after all, a rank-1 update of the form
is positive-definite if
is. The explanation is that the update might be of the form
instead because the denominator can be negative, and in that case there are no guarantees about positive-definiteness.
The SR1 formula has been rediscovered a number of times. Since the denominator can vanish, some authors have suggested that the update be applied only if
,
where
is a small number, e.g.
.[4]
The SR1 update maintains a dense matrix, which can be prohibitive for large problems. Similar to the
L-BFGS method also a limited-memory SR1 (L-SR1) algorithm exists.[5] Instead
of storing the full Hessian approximation, a L-SR1 method only stores the
most recent
pairs
, where
and
is an integer much smaller
than the problem size (
). The limited-memory matrix is
![{\displaystyle B_{k}=B_{0}+J_{k}N_{k}^{-1}J_{k}^{T},\quad J_{k}=Y_{k}-B_{0}S_{k},\quad N_{k}=D_{k}+L_{k}+L_{k}^{T}-S_{k}^{T}B_{0}S_{k}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c1d5a0858995440e26c477b5a3460b3544d0161a)
![{\displaystyle Y_{k}={\begin{bmatrix}y_{k-m}&y_{k-m+1}&\ldots &y_{k-1}\end{bmatrix}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/c66c85fd4590a32a1056630ec49e05493ad15d5b)
![{\displaystyle {\big (}L_{k}{\big )}_{ij}=s_{i-1}^{T}y_{j-1},\quad D_{k}=s_{i-1}^{T}y_{i-1},\quad k-m\leq i\leq k-1}](//wikimedia.org/api/rest_v1/media/math/render/svg/2e187f4cd02bde7b6068baeff18e3d17c789fe22)
Since the update can be indefinite, the L-SR1 algorithm is
suitable for a trust-region strategy. Because of the limited-memory matrix,
the trust-region L-SR1 algorithm scales linearly with the problem size, just like L-BFGS.
Conn, A. R.; Gould, N. I. M.; Toint, Ph. L. (March 1991). "Convergence of quasi-Newton matrices generated by the symmetric rank one update". Mathematical Programming. 50 (1). Springer Berlin/ Heidelberg: 177–195. doi:10.1007/BF01594934. ISSN 0025-5610. S2CID 28028770.
Khalfan, H. Fayez; et al. (1993). "A Theoretical and Experimental Study of the Symmetric Rank-One Update". SIAM Journal on Optimization. 3 (1): 1–24. doi:10.1137/0803001.
Byrd, Richard H.; et al. (1996). "Analysis of a Symmetric Rank-One Trust Region Method". SIAM Journal on Optimization. 6 (4): 1025–1039. doi:10.1137/S1052623493252985.
Nocedal, Jorge; Wright, Stephen J. (1999). Numerical Optimization. Springer. ISBN 0-387-98793-2.
Brust, J.; et al. (2017). "On solving L-SR1 trust-region subproblems". Computational Optimization and Applications. 66: 245–266. doi:10.1007/s10589-016-9868-3.