The global and superlinear convergence of a new nonmonotone MBFGS algorithm on convex objective functions
This article extends two existing papers, namely one by \textit{J. Han} and \textit{G. Liu} [Comput. Optim. Appl. 7, No.~3, 277--289 (1997; Zbl 0876.90072)] and one by \textit{Z. Wei, G. Yu, G. Yuan} and \textit{Z. Lian} [Comput. Optim. Appl. 29, No.~3, 315--332 (2004; Zbl 1070.90089)], these could have been cited properly to shorten the discussed work. As in the latter reference the authors introduce a slightly modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula (MBFGS) for the solution of unconstrained optimization problems, but instead of the more standard Wolfe-type line-search a ``GLL'' line search is used. This on the other hand has been studied with the standard BFGS formula in the primer reference. The steps to prove global convergence are taken from this paper nearly unchanged, so are the lemmas from Wei et al. [loc. cit.] for proving superlinear convergence. The paper ends with numerical comparisons on a test-set where one can see that the difference between the ``nonmonotone'' MBFGS, the Wolfe-MBFGS and also the ``nonmonotone'' standard BFGS is negligible.
Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula
scientific article; zbMATH DE number 5315410