Updating Quasi-Newton Matrices with Limited Storage

Updating Quasi - Newton Matrices With Limited Storage By Jorge Nocedal

We study how to use the BFGS quasi-Newton matrices to precondition minimization methods for problems where the storage is critical. We give an update formula which generates matrices using information from the last m iterations, where m is any number supplied by the user. The quasi-Newton matrix is updated at every iteration by dropping the oldest information and replacing it by the newest info...

Automatic Preconditioning by Limited Memory Quasi-Newton Updating

This paper proposes a preconditioner for the conjugate gradient method (CG) that is designed for solving systems of equations Ax = bi with different right-hand-side vectors or for solving a sequence of slowly varying systems Akx = bk. The preconditioner has the form of a limited memory quasi-Newton matrix and is generated using information from the CG iteration. The automatic preconditioner doe...

On Efficiently Computing the Eigenvalues of Limited-Memory Quasi-Newton Matrices

In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduces the cos...

Representations of quasi-Newton matrices and their use in limited memory methods

We derive compact representations of BFGS and symmetric rank one matrices for optimization These representations allow us to e ciently implement limitedmemory methods for large constrained optimization problems In particular we discuss how to compute projections of limited memory matrices onto subspaces We also present a compact representation of the matrices generated by Broyden s update for s...

Some numerical experiments with variable-storage quasi-Newton algorithms