Dynamic scaling on the limited memory BFGS method

• Updates of the well known LBFGS method are approximately equilibrated.
• The initial diagonal matrix is modified to equilibrate and to approximate the Hessian.
• Numerical results indicate that the proposed scaling strategy is very effective.

This paper describes a limited-memory quasi-Newton method in which the initial inverse Hessian approximation is constructed based on the concept of equilibration of the inverse Hessian matrix. Curvature information about the objective function is stored in the form of a diagonal matrix, and plays the dual role of providing an initial matrix and of equilibrating for limited memory BFGS (LBFGS) iterations. An extensive numerical testing has been performed showing that the diagonal scaling strategy proposed is very effective.