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The quasi-Cauchy relation and diagonal updating. (English) Zbl 1013.90137
Authors’ summary: The quasi-Cauchy (QC) relation is the weak quasi-Newton relation of J. E. Dennis jun. and H. Wolkowicz [SIAM J. Numer. Anal. 30, 1291-1314 (1993; Zbl 0802.65081)] 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.

MSC:
90C30 Nonlinear programming
49M30 Other numerical methods in calculus of variations (MSC2010)
65F30 Other matrix algorithms (MSC2010)
90C53 Methods of quasi-Newton type
Software:
L-BFGS
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