On variable-metric methods for sparse Hessians

The relationship between variable-metric methods derived by norm minimization and those derived by symmetrization of rank-one updates for sparse systems is studied, and an analogue of Dennis’s nonsparse symmetrization formula derived. A new method of using norm minimization to produce a sparse analogue of any nonsparse variable-metric method is proposed. The sparse BFGS generated by this method is tested against the sparse PSB and variable-memory conjugate gradient methods, with computational experience uniformly favoring the sparse BFGS.