On relationships between vector variational inequalities and optimization problems using convexificators on the Hadamard manifold

Nagendra Singh; Sunil Kumar Sharma; Akhlad Iqbal; Shahid Ali; Nagendra Singh; Sunil Kumar Sharma; Akhlad Iqbal; Shahid Ali

doi:10.3934/math.2025259

AIMS Mathematics

2025, Volume 10, Issue 3: 5612-5630. doi: 10.3934/math.2025259

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On relationships between vector variational inequalities and optimization problems using convexificators on the Hadamard manifold

1.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2.
Department of Information Systems College of Computer and Information Sciences, Majmaah University, Majmaah 11952, Saudi Arabia

Received: 10 September 2024 Revised: 26 February 2025 Accepted: 04 March 2025 Published: 13 March 2025
MSC : 47J20, 49J52, 53C22, 58E17

This study extended a fundamental idea about convexificators to the Hadamard manifolds. The mean value theorem for convexificators on the Hadamard manifold was also derived. An important characterization for the bounded convexificators to have $ \partial_{*}^{*} $-geodesic convexity was derived and the monotonicity of the bounded convexificators was explored. Additionally, a convexificator-based vector variational inequality problem on the Hadamard manifold was examined. Furthermore, the necessary and sufficient conditions for vector optimization problems in terms of the Stampacchia and Minty-type partial vector variational inequality problems ($ \partial_{*}^{*} $-VVIPs) were derived.
- geodesic convexity,
- monotonicity,
- $ \partial_{*}^{*} $-VVIP,
- VOP,
- Hadamard manifold
Citation: Nagendra Singh, Sunil Kumar Sharma, Akhlad Iqbal, Shahid Ali. On relationships between vector variational inequalities and optimization problems using convexificators on the Hadamard manifold[J]. AIMS Mathematics, 2025, 10(3): 5612-5630. doi: 10.3934/math.2025259

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Abstract

This study extended a fundamental idea about convexificators to the Hadamard manifolds. The mean value theorem for convexificators on the Hadamard manifold was also derived. An important characterization for the bounded convexificators to have $ \partial_{*}^{*} $-geodesic convexity was derived and the monotonicity of the bounded convexificators was explored. Additionally, a convexificator-based vector variational inequality problem on the Hadamard manifold was examined. Furthermore, the necessary and sufficient conditions for vector optimization problems in terms of the Stampacchia and Minty-type partial vector variational inequality problems ($ \partial_{*}^{*} $-VVIPs) were derived.

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