Optimal conditioning in the convex class of rank two updates

Davidon's new quasi-Newton optimization algorithm selects the new inverse Hessian approximation at each step to be the optimally conditioned member of a certain one-parameter class of rank two updates to the last inverse Hessian approximationH. In this paper we show that virtually the same goals of conditioning can be achieved while restricting to the convex class of updates, which are bounded by the popular DFP and BFGS updates. This suggests the computational testing of alternatives to the optimal conditioning strategy.This research supported by NSF grant 73-03413, contract P04361 of the National Bureau of Economic Research, Cambridge, Massachusetts, and a National Science Foundation Graduate Fellowship, forms a portion of the author's doctoral thesis at Cornell University directed by Professor J.E. Dennis.