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Multiobjective Conjugate Gradient Methods on Riemannian Manifolds

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Abstract

In this paper, we present the multiobjective optimization methods of conjugate gradient on Riemannian manifolds. The concepts of optimality and Wolfe conditions, as well as Zoutendijk’s theorem, are redefined in this setting. We show that under some standard assumptions, a sequence generated by these algorithms converges to a critical Pareto point. This is when the step sizes satisfy the multiobjective Wolfe conditions. In particular, we propose the Fletcher–Reeves, Dai–Yuan, Polak–Ribière–Polyak, and Hestenes–Stiefel parameters and further analyze the convergence behavior of the first two methods and test their performance against the steepest descent method.

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Acknowledgements

The authors are deeply grateful to the editor and anonymous referees, whose patience and numerous detailed comments greatly enhanced the quality of this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    Shahabeddin Najafi & Masoud Hajarian

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  1. Shahabeddin Najafi
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  2. Masoud Hajarian
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Correspondence to Masoud Hajarian.

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Communicated by Sándor Zoltán Németh.

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Najafi, S., Hajarian, M. Multiobjective Conjugate Gradient Methods on Riemannian Manifolds. J Optim Theory Appl 197, 1229–1248 (2023). https://doi.org/10.1007/s10957-023-02224-1

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