Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds

Abstract

In this paper, we extend some results of nonsmooth analysis from Euclidean context to the Riemannian setting. In particular, we discuss the concept and some properties of locally Lipschitz continuous vector fields on Riemannian settings, such as Clarke generalized covariant derivative, upper semicontinuity and Rademacher theorem. We also present a version of Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields. Under mild conditions, we establish the well-definedness and local convergence of the sequence generated by the method in a neighborhood of a singularity. In particular, a local convergence result for semismooth vector fields is presented. Furthermore, under Kantorovich-type assumptions the convergence of the sequence generated by the Newton method to a solution is established, and its uniqueness in a suitable neighborhood of the starting point is verified.

A Global Version of the Newton Method for Finding a Singularity of the Nonsmooth Vector Fields on Riemannian Manifolds

“…Remark 8. According to the Rademacher theorem, see [12,Theorem 3.2], locally Lipschitz continuous vector fields are everywhere differentiable.…”

“…Although the interest in nonsmooth functions in the Riemannian setting has increased; see for example [2,18,21,23,24,25,26,29], only a few studies exist on nonsmooth vector fields in this context; see [21,33]. Recently, [12] proposed and analyzed a version of the Newton method for finding a singularity of a class of locally Lipschitz continuous vector field. For the smooth vector fields, much has already been done, see [1,6,17,19,20,30,36].…”

“…The goal of this paper is to present a global version of the Newton method for finding a singularity of nonsmooth vector fields. Basically, we combine our first algorithm presenting in [12] with the nonmonotone line search strategy. We show that any accumulation point of the iterative sequence is a stationary point of the chosen merit function.…”

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This paper is concerned with an algorithm for finding a singularity of the nonsmooth vector fields. Firstly, we discuss the main results of the Newton method presented in [12] for solving the aforementioned problem. Combining this method with a nonmonotone line search strategy, we then propose a global version of the Newton Method.

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Well-posedness of an interaction model on Riemannian manifolds

“…To this aim we use the concept of Lipschitz continuity of vector fields via parallel transport. While this is a concept widely used in the literature on optimization on manifolds [12,16], it is much less common in the analysis of differential equations on manifolds. Lipschitz continuity by parallel transport enables us to compare tangent vectors in an intrinsic manner.…”