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

TL;DR

This paper extends the concept of convexificators to Hadamard manifolds, deriving key theorems and characterizations, and explores their application to vector variational inequalities and optimization problems on these manifolds.

Contribution

It introduces convexificators on Hadamard manifolds, establishes their properties, and links them to vector variational inequalities and optimization conditions.

Findings

Extended convexificators to Hadamard manifolds.

Derived mean value theorem for convexificators.

Provided necessary and sufficient conditions for vector optimization.

Abstract

An important concept of convexificators has been extended to Hadamard manifolds in this paper. The mean value theorem for convexificators on the Hadamard manifold has also been derived. Monotonicity of the bounded convexificators has been discussed and an important characterization for the bounded convexificators to be $\partial_{*}^{*}$-geodesic convexity has been derived. Furthermore, a vector variational inequalities problem using convexificators on Hadamard manifold has been considered. In addition, the necessary and sufficient conditions for vector optimization problems in terms of Stampacchia and Minty type partial vector variational inequality problem ($\partial_{*}^{*}$-VVIP) have been derived.