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 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.