Metric-based symmetric rank-one updates

Metric-based symmetric rank-one (SR1) updates which are stabilized by a variational relaxation of the quasi-Newton relation are examined. This investigation reveals an interesting and surprising connection to the origin of quasi-Newton methods as first formulated by Davidon. An extended version of Davidon's original direct prediction SR1 update is shown to be self-complementary and to possess a finite termination property on quadratics, and limited-memory versions of the update are shown to be globally convergent. Variants of this update are tested numerically and compared with several other metric-based SR1 variants and the Broyden–Fletcher–Goldfarb–Shanno update. Finally, metric-based stabilizations of the SR1 update are critiqued in general, and a promising new model-based strategy recently developed is briefly described.