An Optimal Positive Definite Update for Sparse Hessian Matrices

A Hessian update is described that preserves sparsity and positive definiteness and satisfies a minimal change property. The update reduces to the BFGS update in the dense case and generalises a recent result in [SIAM J. Nnmer. Anal., 26 (1989), pp. 727–739] relating to the Byrd and Nocedal measure function. A surprising outcome is that a sparsity projection of the inverse Hessian plays a major role. It is shown that the Hessian itself can be recovered from this information under mild assumptions. The update is computed by solving a concave programming problem derived by using the Wolfe dual. The Hessian of the dual is important and plays a similar role to the matrix Q that arises in the sparse PSB update of Toint [Math. Comp., 31 (1977), pp. 954–961]. This matrix is shown to satisfy the same structural and definiteness conditions as Toint's matrix. The update has been implemented for tridiagonal systems and some numerical experiments are described. These experiments indicate that there is potential for a significant reduction in the number of quasi-Newton iterations, but that more development is needed to obtain an efficient implementation. Solution of the variational problem by primal methods is also discussed and provides an interesting application of generalized elimination. The possibility of instability and nonexistence of a positive definite update raised by Sorensen [Math. Programming Study, 18 (1982), pp. 135–159] is still a difficulty and some remedies are discussed.