Dennis-Moré Condition for Set-Valued Vector Fields and the Superlinear Convergence of Broyden Updates in Riemannian Manifolds

This paper deals with the quasi-Newton type scheme for solving generalized equations involving set-valued vector fields on Riemannian manifolds. We establish some conditions ensuring the superlinear convergence for the iterative sequence which approximates a solution of the generalized equations. Such conditions can be viewed as an extension of the classical theorem of J. E. Dennis and J. J. Moré [see: A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Computation 28/126 (1974) 549–560] as well as the Riemannian Dennis-Moré condition established by K. A. Gallivan, C. Qi and P.-A. Absil [A Riemannian Dennis-Moré Condition, in: High-Performance Scientific Computing: Algorithms and Applications, M. W. Berry et al. (eds.), Springer, London (2012) 281–293]. Furthermore, we also apply these results to consider the convergence of a Broyden-type update for the problem of solving generalized equations in Riemannian context. Our results are new even for classical equations defined by single-valued vector fields.