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Gauss–Newton method for convex composite optimizations on Riemannian manifolds

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Abstract

A notion of quasi-regularity is extended for the inclusion problem \({F(p)\in C}\) , where F is a differentiable mapping from a Riemannian manifold M to \({\mathbb R^n}\) . When C is the set of minimum points of a convex real-valued function h on \({\mathbb R^n}\) and DF satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function h ◦ F on Riemannian manifold. Two applications are provided: one is for the case of regularities on Riemannian manifolds and the other is for the case when C is a cone and DF(p 0)(·) − C is surjective. In particular, the results obtained in this paper extend the corresponding one in Wang et al. (Taiwanese J Math 13:633–656, 2009).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Technology, Hangzhou, 310032, People’s Republic of China

    Jin-Hua Wang

  2. Department of Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan

    Jen-Chih Yao

  3. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Chong Li

  4. Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

    Chong Li

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  1. Jin-Hua Wang
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  2. Jen-Chih Yao
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  3. Chong Li
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Correspondence to Jen-Chih Yao.

Additional information

Jin-Hua Wang was supported in part by the National Natural Science Foundation of China (grant 11001241).

Jen-Chih Yao was supported in part by the Grant NSC 99-2115-M-110-004-MY3.

Chong Li was supported in part by the National Natural Science Foundation of China (grant 10731060).

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Wang, JH., Yao, JC. & Li, C. Gauss–Newton method for convex composite optimizations on Riemannian manifolds. J Glob Optim 53, 5–28 (2012). https://doi.org/10.1007/s10898-010-9638-1

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