Measures for Symmetric Rank-One Updates

Measures of deviation of a symmetric positive deenite 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 diierence of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. The measure deened by the problem-maximize the determinant subject to a bound of 1 on the largest eigenvalue-yields the SR1 update. The measure (A) = 1(A) det(A) 1 n yields the optimally conditioned, sized, symmetric rank-one updates, 1, 2]. The volume considerations also suggest a `correction' for the initial stepsize for these sized updates. It is then shown that the-optimal updates, as well as the Oren-Luenberger self-scaling updates 3], are all optimal updates for the measure, thè 2 condition number. Moreover, all four sized updates result in the same largest (and smallest) 'scaled' eigenvalue and corresponding eigenvector. In fact, the inverse-sized BFGS is the mean of the-optimal updates, while the inverse of the sized DFP is the mean of the inverses of the-optimal updates. The diierence between these four updates is determined by the middle n?2 scaled eigenvalues. The measure also provides a natural Broyden class replacement for the SR1 when it is not positive deenite.