Monotone and Accretive Vector Fields on Riemannian Manifolds

Abstract

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings.We also establish the equivalence between the strong convexity of convex functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization problem of convex functions on Riemannian manifolds.

Resolvents of Set-Valued Monotone Vector Fields in Hadamard Manifolds

“…The concept of accretivity of vector fields on Riemannian manifolds was introduced in [48] and previously studied in other nonlinear metric spaces in [23,40,44]. In [48] the authors proved that in the setting of Hadamard manifolds the notions of monotonicity and accretivity are equivalent.…”

“…The study of accretive and monotone type operators in metric spaces has been the focus of many authors in the last decades; see, for example, [10,12,13,22,23,27,28,30,32,34,40,44]. It is worth mentioning that, as happens in the case of Hilbert spaces, in Hadamard manifolds, which are Riemannian manifolds of nonpositive sectional curvature, the class of maximal monotone vector fields coincides with the class of m-accretive vector fields, see [48].…”

Proximal Point Algorithms on Hadamard Manifolds: Linear Convergence and Finite Termination

“…Then, by [58], V y is 1-strongly maximal monotone. Furthermore, for any λ > 0 and any monotone vector field A on M , the multivalued vector field A λ,y defined by…”

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

“…There are some advantages for a generalization of optimization methods from Euclidean spaces to Riemannian manifolds, because nonconvex and nonsmooth constrained optimization problems can be seen as convex and smooth unconstrained optimization problems from the Riemannian geometry point of view; see, for example, [15], [11], [12]. Nemeth [10] and Wang et al [16] studied monotone and accretive vector fields on Riemannian manifolds. Li et al [5] extended maximal monotone vector fields from Banach spaces to Hadamard manifolds (simply connected complete Riemannian manifold with nonpositive sectional curvature).…”