Characterizations of locally Lipschitz geodesic pseudolinear functions on Hadamard manifolds

In this paper, using the properties of Clarke subdifferential we derive some characterizations of locally Lipschitz geodesic pseudolinear functions on Hadamard manifolds. Moreover, employing these characterizations we derive some properties of the solution sets of pseudolinear programming problems on the Hadamard manifold. The results established in this paper generalize and extend several known results in the literature. Non-trivial examples have been formulated to demonstrate these results. Additionally, we discuss a practical application, namely, the Karcher mean problem, and show that under certain assumptions, the objective function of this problem becomes geodesic pseudolinear. Moreover, to demonstrate the practical relevance of our outcomes, we verify their applicability to this specific form of the Karcher mean problem.