MEASURES FOR SYMMETRICAL RANK-ONE UPDATES

Measures of deviation of a symmetric positive definite matrix from the identity are derived. They give rise to symmetric rank-one, SR1, type updates; The measures are motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the curre nt and updated quadratic models in quasi-Newton methods. The measure d efined by the problem-maximize the determinant subject to a bound of 1 on the largest eigenvalue-yields the SR1 update. The measure sigma(A) = lambda(1)(A)/det(A)(1)/(n) yields the optimally conditioned, sized, symmetric rank-one updates. The volume considerations also suggest a 'correction' for the initial stepsize for these sized updates. It is t hen shown that the sigma-optimal updates, as well as the Oren-Luenberg er self-scaling updates, are all optimal updates for the kappa measure , the l(2) condition number. Moreover, all four sized updates result i n the same largest (and smallest) 'scaled' eigenvalue and correspondin g eigenvector. In fact, the inverse-sized BFGS is the mean of the sigm a-optimal updates, while the inverse of the sized DFP is the mean of t he inverses of the sigma-optimal updates. The difference between these four updates is determined by the middle n - 2 scaled eigenvalues. Th e kappa measure also provides a natural Broyden class replacement for the SR1 when it is not positive definite.