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<p>Optimization problems on curved spaces such as Hadamard manifolds arise in machine learning, computer vision, and data analysis. Motivated by these applications, this paper studies the interplay between nonsmooth vector variational inequality problems (NVVIP), their Minty-type counterparts (MNVVIP), and nonsmooth vector optimization problems (NVOP) on Hadamard manifolds. Using vector-valued bifunctions as the main tool, we develop a framework for generalized geodesic convexity, introducing geodesic <i>h</i>-convexity, <i>h</i>-pseudoconvexity, and <i>h</i>-quasiconvexity for vector-valued functions, and illustrate their structure through a non-trivial example. Relying on bifunctional methods and generalized convexity techniques, we prove uniqueness results for NVVIP and establish characterizations linking NVVIP with MNVVIP. We also derive conditions under which solutions of NVVIP, MNVVIP, and NVOP coincide, providing a unified theoretical perspective. These results not only extend the theory of vector variational inequalities and optimization on manifolds but also offer foundational tools for applications in optimization over non-Euclidean spaces.</p>

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Relations Between Nonsmooth Vector Variational Inequalities and Nonsmooth Vector Optimization Problems on Hadamard Manifold in Terms of Bifunction

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Nagendra Singh,
  • Akhlad Iqbal,
  • Shahid Ali

摘要

Optimization problems on curved spaces such as Hadamard manifolds arise in machine learning, computer vision, and data analysis. Motivated by these applications, this paper studies the interplay between nonsmooth vector variational inequality problems (NVVIP), their Minty-type counterparts (MNVVIP), and nonsmooth vector optimization problems (NVOP) on Hadamard manifolds. Using vector-valued bifunctions as the main tool, we develop a framework for generalized geodesic convexity, introducing geodesic h-convexity, h-pseudoconvexity, and h-quasiconvexity for vector-valued functions, and illustrate their structure through a non-trivial example. Relying on bifunctional methods and generalized convexity techniques, we prove uniqueness results for NVVIP and establish characterizations linking NVVIP with MNVVIP. We also derive conditions under which solutions of NVVIP, MNVVIP, and NVOP coincide, providing a unified theoretical perspective. These results not only extend the theory of vector variational inequalities and optimization on manifolds but also offer foundational tools for applications in optimization over non-Euclidean spaces.