Symmetric rank-one

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.{{cite journal|last1=Conn|first1=A. R.|last2=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 |issn=0025-5610|pages=177–195|volume=50|number=1|doi=10.1007/BF01594934|s2cid=28028770 }}{{cite journal |last1=Khalfan |first1=H. Fayez |first2=R. H. |last2=Byrd |first3=R. B. |last3=Schnabel |display-authors=1 |year=1993 |title=A Theoretical and Experimental Study of the Symmetric Rank-One Update |journal=SIAM Journal on Optimization |volume=3 |issue=1 |pages=1–24 |doi=10.1137/0803001 }} The SR1 method has computational advantages for sparse or partially separable problems.{{cite journal |last1=Byrd |first1=Richard H. |first2=Humaid Fayez |last2=Khalfan |first3=Robert B. |last3=Schnabel |display-authors=1 |year=1996 |title=Analysis of a Symmetric Rank-One Trust Region Method |journal=SIAM Journal on Optimization |volume=6 |issue=4 |pages=1025–1039 |doi=10.1137/S1052623493252985 }}

A twice continuously differentiable function $x\; \backslash mapsto\; f(x)$ has a gradient ($\backslash nabla\; f$) and Hessian matrix $B$: The function $f$ has an expansion as a Taylor series at $x\_0$, which can be truncated

::$f(x\_0+\backslash Delta\; x)\; \backslash approx\; f(x\_0)+\backslash nabla\; f(x\_0)^T\; \backslash Delta\; x+\backslash frac\{1\}\{2\}\; \backslash Delta\; x^T\; \{B\}\; \backslash Delta\; x$;

its gradient has a Taylor-series approximation also

::$\backslash nabla\; f(x\_0+\backslash Delta\; x)\; \backslash approx\; \backslash nabla\; f(x\_0)+B\; \backslash Delta\; x$,

which is used to update $B$. The above secant-equation need not have a unique solution $B$.

The SR1 formula computes (via an update of rank 1) the symmetric solution that is closest{{explain|date=July 2022}} to the current approximate-value $B\_k$:

::$B\_\{k+1\}=B\_\{k\}+\backslash frac\; \{(y\_k-B\_k\; \backslash Delta\; x\_k)\; (y\_k-B\_k\; \backslash Delta\; x\_k)^T\}\{(y\_k-B\_k\; \backslash Delta\; x\_k)^T\; \backslash Delta\; x\_k\}$,

where

::$y\_k=\backslash nabla\; f(x\_k+\backslash Delta\; x\_k)-\backslash nabla\; f(x\_k)$.

The corresponding update to the approximate inverse-Hessian $H\_k=B\_k^\{-1\}$ is

::$H\_\{k+1\}=H\_\{k\}+\backslash frac\; \{(\backslash Delta\; x\_k-H\_k\; y\_k)(\backslash Delta\; x\_k-H\_k\; y\_k)^T\}\{(\backslash Delta\; x\_k-H\_k\; y\_k)^T\; y\_k\}$.

One might wonder why positive-definiteness is not preserved — after all, a rank-1 update of the form $B\_\{k+1\}\; =\; B\_k\; +\; vv^T$ is positive-definite if $B\_k$ is. The explanation is that the update might be of the form $B\_\{k+1\}\; =\; B\_k\; -\; vv^T$ 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

::$|\backslash Delta\; x\_k^T\; (y\_k-B\_k\; \backslash Delta\; x\_k)|\backslash geq\; r\; \backslash |\backslash Delta\; x\_k\backslash |\backslash cdot\; \backslash |y\_k-B\_k\; \backslash Delta\; x\_k\backslash |$,

where $r\backslash in(0,1)$ is a small number, e.g. $10^\{-8\}$.{{cite book |last1=Nocedal |first1=Jorge |last2=Wright |first2=Stephen J. |year=1999 |title=Numerical Optimization |publisher=Springer |isbn=0-387-98793-2 }}