- SR1 formula
The Symmetric Rank 1 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 thesecant method for a multidimensional problem.This update maintains the symmetry of the matrix but does not guarantee the update to be apositive definite matrix .For this reason it is the method of choice for indefinite problems.Given a function , its
gradient (), andHessian matrix , theTaylor series is:::,and theTaylor series of the gradient itself:::,is used to update . Equation above (secant equation) can admit an infinite number of solutions to . The SR1 formula finds a solution of rank 1 that is symmetric and closest to the current approximate value of :::,where::.The corresponding update to the inverse Hessian approximation is given by:::.The derivation is simple, and the SR1 formula has been rediscovered a number of times.The main drawback is that the denominator can vanish. It is therefore a good idea to apply the SR1 update only if::,where is a small number, e.g. .
ee also
*
Quasi-Newton method
*Newton's method in optimization
* Broyden-Fletcher-Goldfarb-Shanno (BFGS) method
* L-BFGS methodReferences
* Nocedal, Jorge & Wright, Stephen J. (1999). Numerical Optimization. Springer-Verlag. ISBN 0-387-98793-2.
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