Measures for Symmetric 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 current and updated quadratic models in quasi-Newton methods. The measure defined by the problem—maximize the determinant subject to a bound of 1 on the largest eigenvalue—yields the SR1 update. The measure σ(A) = λ1(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 then shown that the σ-optimal updates, as well as the Oren-Luenberger self-scaling updates, are all optimal updates for the κ measure, the l2 condition number. Moreover, all four sized updates result in the same largest (and smallest) ‘scaled’ eigenvalue and corresponding eigenvector. In fact, the inversesized BFGS is the mean of the σ-optimal updates, while the inverse of the sized DFP is the mean of the inverses of the σ-optimal updates. The difference between these four updates is determined by the middle n − 2 scaled eigenvalues. The κ measure also provides a natural Broyden class replacement for the SR1 when it is not positive definite.