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 add
ed restriction that full matrices are replaced by diagonal matrices. This r
elation 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 d
iagonal updated matrix with hereditary positive definiteness is used to pre
condition Cauchy's steepest-descent direction. The resulting QC algorithm i
s shown to be significantly accelerated.
In the concluding section, the following topics are briefly discussed: addi
tional variational principles, use of diagonal updates within other optimiz
ation algorithms together with some further numerical experience (summarize
d in an appendix), and an interesting connection between QC-diagonal updati
ng and trust-region techniques.