Relations between nonsmooth vector variational inequalities and nonsmooth vector optimization problems on Hadamard manifold in terms of bifunction

TL;DR

This paper explores the relationships between nonsmooth vector variational inequalities and nonsmooth vector optimization problems on Hadamard manifolds, introducing new convexity concepts and characterizing solutions using bifunctions.

Contribution

It introduces generalized geodesic convexity concepts on Hadamard manifolds and characterizes solutions of variational inequalities and optimization problems in this context.

Findings

Defined geodesic $h$-convexity, $h$-pseudoconvexity, and $h$-quasiconvexity for vector functions.

Proved the uniqueness of solutions for NVVIP on Hadamard manifolds.

Established relationships among solutions of NVOP, NVVIP, and MNVVIP.

Abstract

In this paper, we discuss the concepts of bifunction and geodesic convexity for vector valued functions on Hadamard manifold. The Hadamard manifold is a particular type of Riemannian manifold with non-positive sectional curvature. Using bifunction, we introduce a definition of generalized geodesic convexity in the context of the Hadamard manifold. To support the definition, we construct a non-trivial example that demonstrates the property of geodesic convexity on Hadamard manifold. Additionally, we define the geodesic $h$-convexity, geodesic $h$-pseudoconvexity and geodesic $h$-quasiconvexity for vector valued function using bifunction and study their several properties. Furthermore, we demonstrate the uniqueness of the solution for nonsmooth vector variational inequality problem (NVVIP) and prove the characterization property for the solution of NVVIP and the Minty type NVVIP (MNVVIP) on Hadamard manifold in terms of bifunction. Afterward, we consider a nonsmooth vector optimization problem (NVOP) and investigate the relationships among the solutions of NVOP, NVVIP, and MNVVIP.