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A. Belyakov, F. Aragon Artacho, A. Dontchev, M. Lopez:
"Local convergence of quasi-Newton methods under metric regularity";
Computational Optimization and Applications, Oct. 2013 (2013), 23 pages.

We consider quasi-Newton methods for generalized equations in Banach
spaces under metric regularity and give a sufficient condition for q-linear convergence.
Then we show that the well-known Broyden update satisfies this sufficient
condition in Hilbert spaces. We also establish various modes of q-superlinear convergence
of the Broyden update under strong metric subregularity, metric regularity
and strong metric regularity. In particular, we show that the Broyden update applied
to a generalized equation in Hilbert spaces satisfies the Dennis-Moré condition for
q-superlinear convergence. Simple numerical examples illustrate the results.

Generalized equation · Quasi-Newton method · Broyden update · Strong metric subregularity · Metric regularity · Strong metric regularity · q-Superlinear convergence

http://dx.doi.org/10.1007/s10589-013-9615-y