The Quasi-Cauchy Relation and Diagonal Updating

The quasi-Cauchy (QC) relation is the weak quasi-Newton relation of Dennis and Wolkowicz [SIAM J. Numer. Anal., 30 (1993), pp. 1291–1314] with the added restriction that full matrices are replaced by diagonal matrices. This relation is justified and explored and, in particular, two basic variational techniques for updating diagonal matrices that satisfy it are formulated. For purposes of illustration, a numerical experiment is described where a diagonal updated matrix with hereditary positive definiteness is used to precondition Cauchy’s steepest-descent direction. The resulting QC algorithm is shown to be significantly accelerated. In the concluding section, the following topics are briefly discussed: additional variational principles, use of diagonal updates within other optimization algorithms together with some further numerical experience (summarized in an appendix), and an interesting connection between QC-diagonal updating and trust-region techniques.