Strongly geodesic preinvexity and strongly invariant -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 Condition C (The Condition C is defined in Remark 1 of this article), defined by Barani and Pouryayevali [J. Math. Anal. Appl. 328 (2007) 767-779]. 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.