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Convergence of Inexact Steepest Descent Algorithm for Multiobjective Optimizations on Riemannian Manifolds Without Curvature Constraints

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Abstract

We study the issue of convergence for inexact steepest descent algorithm (employing general step sizes) for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity, local/global convergence results are established. Furthermore, without the assumption of the local convexity/quasi-convexity, but under an error bound-like condition, local/global convergence results and convergence rate estimates are presented, which are new even in the linear space setting. Our results improve/extend the corresponding ones in (Wang et al. in SIAM J Optim 31(1):172–199, 2021) for scalar optimization problems on Riemannian manifolds to multiobjective ones. Finally, for the special case when the inexact steepest descent algorithm employing Armijo rule, our results improve/extend the corresponding ones in (Ferreira et al. in J Optim Theory Appl 184:507–533, 2020) by relaxing curvature constraints.

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Acknowledgements

The authors are indebted to the handling editor and two anonymous referees for their very helpful remarks and comments, which allowed us to improve the original presentation. Research of the first author is supported in part by the National Natural Science Foundation of China (Grant 12161017) and Guizhou Provincial Natural Science Foundation of China (Grant ZK[2022]110). Research of the second author was supported in part by the National Natural Science Foundation of China (Grant 12171131). Research of the third author was supported in part by the National Natural Science Foundation of China (Grant 11971429).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, People’s Republic of China

    X. M. Wang

  2. Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, People’s Republic of China

    J. H. Wang

  3. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310058, People’s Republic of China

    C. Li

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  1. X. M. Wang
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  2. J. H. Wang
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  3. C. Li
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Correspondence to J. H. Wang.

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Communicated by Sándor Zoltán Németh.

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Wang, X.M., Wang, J.H. & Li, C. Convergence of Inexact Steepest Descent Algorithm for Multiobjective Optimizations on Riemannian Manifolds Without Curvature Constraints. J Optim Theory Appl 198, 187–214 (2023). https://doi.org/10.1007/s10957-023-02235-y

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