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Journal of Convex Analysis 29 (2022), No. 3, 669--701
Copyright Heldermann Verlag 2022

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

Samir Adly
Laboratoire XLIM, Université de Limoges, France

Huynh Van Ngai
Dept. of Mathematics and Statistics, Quy Nhon University, Quy Nhon, Binh Dinh, Vietnam

Van Vu Nguyen
Dept. of Mathematics and Statistics, Quy Nhon University, Quy Nhon, Binh Dinh, Vietnam

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.

Keywords: Variational inclusion, point-to-set vector fields, quasi-Newton methods, Riemannian manifold, Dennis-Moré condition, superlinear convergence.

MSC: 65J99, 65K15, 58C06, 47H04.

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