In this paper, we propose nonlinear conjugate gradient methods for vector optimization on Riemannian manifolds. The concepts of Wolfe and Zoutendjik conditions are extended to Riemannian manifolds. Specifically, the existence of intervals of step sizes that satisfy the Wolfe conditions is established. The convergence analysis covers the vector extensions of the Fletcher–Reeves, conjugate descent, and Dai–Yuan parameters. Under some assumptions, we prove that the sequence obtained by the proposed algorithm can converge to a Pareto stationary point. Moreover, several other choices of the parameter are discussed. Numerical experiments illustrating the practical behavior of the methods are presented.

中文翻译：

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用于黎曼流形上具有回缩和矢量传递的向量优化的非线性共轭梯度方法


在本文中，我们提出了用于黎曼流形向量优化的非线性共轭梯度方法。Wolfe 和 Zoutendjik 条件的概念扩展到黎曼流形。具体来说，确定了满足 Wolfe 条件的步长区间的存在。收敛分析涵盖 Fletcher-Reeves 的向量扩展、共轭下降和 Dai-Yuan 参数。在一些假设下，我们证明了所提出的算法得到的序列可以收敛到帕累托静止点。此外，还讨论了该参数的其他几种选择。提出了说明这些方法实际行为的数值实验。