标题： 基于双函数的Hadamard流形上非光滑向量变分不等式与非光滑向量优化问题之间的关系 显示英文标题

标题： Relations between nonsmooth vector variational inequalities and nonsmooth vector optimization problems on Hadamard manifold in terms of bifunction

摘要： 在本文中，我们讨论了Hadamard流形上向量值函数的双函数和测地凸性的概念。 Hadamard流形是一种具有非正截面曲率的黎曼流形的特殊类型。 使用双函数，我们在Hadamard流形的背景下引入了广义测地凸性的定义。 为了支持该定义，我们构造了一个非平凡的例子，以展示Hadamard流形上的测地凸性性质。 此外，我们利用双函数定义了向量值函数的测地$h$-凸性、测地$h$-伪凸性和测地$h$-拟凸性，并研究了它们的几个性质。 此外，我们展示了非光滑向量变分不等式问题（NVVIP）解的唯一性，并证明了在双函数的基础上，Hadamard流形上NVVIP和Minty型NVVIP（MNVVIP）解的表征性质。 随后，我们考虑了一个非光滑向量优化问题（NVOP），并研究了NVOP、NVVIP和MNVVIP解之间的关系。 显示英文摘要

摘要： In this paper, we discuss the concepts of bifunction and geodesic convexity for vector valued functions on Hadamard manifold. The Hadamard manifold is a particular type of Riemannian manifold with non-positive sectional curvature. Using bifunction, we introduce a definition of generalized geodesic convexity in the context of the Hadamard manifold. To support the definition, we construct a non-trivial example that demonstrates the property of geodesic convexity on Hadamard manifold. Additionally, we define the geodesic $h$-convexity, geodesic $h$-pseudoconvexity and geodesic $h$-quasiconvexity for vector valued function using bifunction and study their several properties. Furthermore, we demonstrate the uniqueness of the solution for nonsmooth vector variational inequality problem (NVVIP) and prove the characterization property for the solution of NVVIP and the Minty type NVVIP (MNVVIP) on Hadamard manifold in terms of bifunction. Afterward, we consider a nonsmooth vector optimization problem (NVOP) and investigate the relationships among the solutions of NVOP, NVVIP, and MNVVIP.