Strongly geodesic preinvexity and Strongly Invariant η-Monotonicity on Riemannian Manifolds and its Application

This paper introduces the concepts of strongly geodesic preinvexity, strongly $\eta$-invexity of order $m$, and strongly invariant $\eta$-monotonicity of order $m$ on Riemannian manifolds. Additionally, it discusses an important characterization of these functions under a condition, known as $\text{{\bf Condition C}}^{\dagger}$, defined by Barani \cite{Poury1}. The paper provides various non-trivial examples to support these definitions. Furthermore, it presents a significant characterization of strict $\eta$-minimizers (or $\eta$-minimizers) of order $m$ for multi-objective optimization problems and a solution to the vector variational-like inequality problem.