SIZING AND LEAST-CHANGE SECANT METHODS

Oren and Luenberger introduced in 1974 a strategy for replacing Hessia n approximations by their scalar multiples and then performing quasi-N ewton updates, generally least-change secant updates such as the BFGS or DFP updates [Oren and Luenberger, Management Sci., 20 (1974), pp. 8 45-862]. In this paper, the function omega(A) = (trace(A)/n)/det(A)1/n ) is shown to be a measure of change with a direct connection to the O ren-Luenberger strategy. This measure is interesting because it is rel ated to the l2 Condition number, but it takes all the eigenvalues of A into account rather than just the extremes. If the class of possible updates is restricted to the Broyden class, i.e., scalar premultiples are not allowed, then the optimal update depends on the dimension of t he problem. It may, or may not, be in the convex class, but it becomes the BFGS update as the dimension increases. This seems to be yet anot her explanation for why the optimally conditioned updates are not sign ificantly better than the BFGS update. The theory results in several n ew interesting updates including a self-scaling, hereditarily positive definite, update in the Broyden class which is not necessarily in the convex class. This update, in conjunction with the Oren-Luenberger sc aling strategy at the first iteration only, was consistently the best in numerical tests.