Rank Modifications of Semi-definite Matrices with Applications to Secant Updates

The BFGS and DFP updates are perhaps the most successful Hessian and inverse Hessian approximations respectively for unconstrained minimization problems. This paper describes these methods in terms of two successive steps: rank reduction and rank restoration. From rank subtractivity and a powerful spectral result, the first step must necessarily result in a positive semidefinite matrix; and the second step is designed to restore positive definiteness. The goal of the research is to better understand the workings of the BFGS and DFP updates to see how they may be modified and yet retain their basic rank and spectral characteristics. The class of BFGS and DFP updates is generalized both in terms of choices for update vectors and rank of the modifications in the formulas. The rank restoration step generalizes naturally to rectangular matrices.