This article deals with the classes of approximate Minty- and Stampacchia-type vector variational inequalities on Hadamard manifolds and a class of nonsmooth interval-valued vector optimization problems. By using the Clarke subdifferentials, we define a new class of functions on Hadamard manifolds, namely, the geodesic $LU$-approximately convex functions. Under geodesic $LU$-approximate convexity hypothesis, we derive the relationship between the solutions of these approximate vector variational inequalities and nonsmooth interval-valued vector optimization problems. This paper extends and generalizes some existing results in the literature.

In this paper, we consider a class of interval-valued multiobjective optimization problems (in short, (IVMOP)) and formulate an associated multiobjective optimization problem, referred to as (MOP). We establish that the Pareto optimal solution of the associated (MOP) is an effective solution of (IVMOP). Using this characteristic of the associated (MOP), we introduce a variant of Newton's algorithm for the considered (IVMOP). The proposed algorithm exhibits superlinear convergence to a locally effective solution of (IVMOP), provided the objective function of (IVMOP) is twice generalized Hukuhara differentiable and locally strongly convex. Furthermore, if the second-order generalized Hukuhara partial derivatives of the objective function of (IVMOP) are generalized Hukuhara Lipschitz continuous, the rate of convergence is quadratic. We provide a suitable numerical example to illustrate the developed methodology. Moreover, we employ the proposed algorithm to solve a real-life portfolio optimization problem.