METRIC-BASED SYMMETRICAL RANK-ONE UPDATES

Metric-based SR1 updates which are stabilized by a variational relaxat ion of the quasi-Newton relation are examined. This investigation reve als an interesting and surprising connection to the origin of quasi-Ne wton methods as first formulated by Davidon [1]. An extended version o f Davidon's original direct prediction SR1 update is shown to be self- complementary and to possess a finite termination property on quadrati cs, and limited-memory versions of the update are shown to be globally convergent. Variants of this update are tested numerically and compar ed to several other metric-based SR1 variants and the BFGS update. Fin ally, metric-based stabilizations of the SR1 update are critiqued in g eneral, and a promising new model-based strategy recently developed is briefly described.