Abstract
In this article, we introduce a new class of functions called r-invexity and geodesic r-preinvexity functions on a Riemannian manifolds. Further, we establish the relationships between r-invexity and geodesic r-preinvexity on Riemannian manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under geodesic r-preinvexity on Riemannian manifolds. In the end, a mean value inequality is extended to a Cartan-Hadamard manifold. The results presented in this paper extend and generalize the results that have appeared in the literature. MSC:58E17, 90C26.
Highlights
Convexity is one of the most frequently used hypotheses in optimization theory
Motivated by work of Barani and Pouryayevali [ ] and Antczak [, ], we introduce the concept of geodesic r-preinvex functions and r-invex functions on Riemannian manifolds, which is a generalization of preinvexity as defined in [, ]
Definition . [ ] Let M be an n-dimensional Riemannian manifold and S be an open subset of M which is geodesic invex set with respect to η : M × M → TM
Summary
Convexity is one of the most frequently used hypotheses in optimization theory. It is well known that a local minimum is a global minimum for a convex function. Motivated by work of Barani and Pouryayevali [ ] and Antczak [ , ], we introduce the concept of geodesic r-preinvex functions and r-invex functions on Riemannian manifolds, which is a generalization of preinvexity as defined in [ , ].
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