Extended Newton’s Method for Mappings on Riemannian Manifolds With Values in a Cone

Abstract

Abstract. Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When Df satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton's method. As applications, we also obtain Kantorovich's type theorem, Smale's type theorem under the γ-condition and an extension of the theory of Smale's approximate zeros.

Gauss–Newton method for convex composite optimizations on Riemannian manifolds

“…Furthermore, two applications to special cases are provided in section 4: one is for the case of regularities on Riemannian manifolds and the other is for the case when C is a cone and DF( p 0 )(·) − C is surjective. In particular, the results obtained in section 4 extend the corresponding one in [48].…”

“…We specialize our results in two important cases: the classical Lipschitz condition and the γ -condition, which have been used extensively in the study of convergence of Newton's method both in Banach spaces and Riemannian manifolds, see for example [25][26][27]48,49,[51][52][53][54].…”

“…Let F : M → R n be a C κ -mapping. Following [27] (see also [9,48]), we define inductively the derivative of order k for F. Recall that ∇ is the Levi-Civita connection on M. Let C κ (T M) denote the set of all the C κ -vector fields on M and C κ (M) the set of all C κ -mappings from M to R, respectively.…”

“…Noting that the mapping F is a special example of the section, so Proposition 3.1 below is a direct consequence of [27,Proposition 5.1] (see also [48] for a direct proof). 0 in B( p 0 , r ).…”

“…Recall that in [48], Robinson's generalized Newton's method is extended to solve the inclusion problem on Riemannian manifolds, i.e., finding a point p * ∈ M such that F( p * ) ∈ C, (4.26) where C is a nonempty closed convex cone in R n and F is a C 1 mapping from a manifold M onto R n . The extended Newton's method for the inclusion problem (4.26) is defined as follows.…”