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Convergence analysis of inexact proximal point algorithms on Hadamard manifolds

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Abstract

Inexact proximal point methods are extended to find singular points for multivalued vector fields on Hadamard manifolds. Convergence criteria are established under some mild conditions. In particular, in the case of proximal point algorithm, that is, \(\varepsilon _n=0\) for each \(n\), our results improve sharply the corresponding results in Li et al. (2009). Applications to optimization problems, variational inequality problems and gradient methods are also given.

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Authors and Affiliations

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

    Jinhua Wang

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

    Jinhua Wang

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

    Chong Li

  4. Departamento de Análisis Matematico, Universidad de Sevilla, Apdo. 1160, 41080 , Sevilla, Spain

    Genaro Lopez

  5. Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung , 80702, Taiwan

    Jen-Chih Yao

  6. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah , 21589, Saudi Arabia

    Jen-Chih Yao

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  1. Jinhua Wang
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  2. Chong Li
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  3. Genaro Lopez
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  4. Jen-Chih Yao
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Correspondence to Chong Li.

Additional information

The first author was partially supported by the National Natural Science Foundation of China (Grant 11001241; 11371325) and by Zhejiang Provincial Natural Science Foundation of China (Grant LY13A010011). The second author was partially by the National Natural Science Foundation of China (Grant 11171300). The third author was partially supported by DGES, Grant MTM2012-34847-C02-01 and Junta de Andalucía, Grant P08-FQM-03453. The last author was partially supported by a grant from NSC of Taiwan (NSC 102-2115-M-037-002-MY3).

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Wang, J., Li, C., Lopez, G. et al. Convergence analysis of inexact proximal point algorithms on Hadamard manifolds. J Glob Optim 61, 553–573 (2015). https://doi.org/10.1007/s10898-014-0182-2

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