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
samir.adly@unilim.fr
Huynh Van Ngai
Dept. of Mathematics and Statistics, Quy Nhon University, Quy Nhon, Binh Dinh, Vietnam
ngaivn@yahoo.com
Van Vu Nguyen
Dept. of Mathematics and Statistics, Quy Nhon University, Quy Nhon, Binh Dinh, Vietnam
nguyenvanvu@qnu.edu.vn
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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