Newton method with feasible inexact projections for constrained equations and nonsmooth Newton method in Riemannian manifolds - Oliveira, Fabiana Rodrigues de (2019)
In this thesis, we will study three versions of the Newton method for solving problems in two contexts, namely Euclidean and Riemannian. In the Euclidean context, we will present the Newton method with feasible inexact projections for solving generalized equations subject to a set of constraints. Under local assumptions, the linear or superlinear convergence of a sequence generated by the proposed method is established. Next, a version of the inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations is presented. Using suitable assumptions, the linear or superlinear convergence of a sequence generated by the method is proved. Furthermore, to illustrate the practical behavior of the proposed method, some numerical experiments are reported. Under another perspective, the last version of the Newton method to be investigated is an extension of the nonsmooth Newton method itself from the Euclidean context to the Riemannian, objecting to find a singularity of a special class of locally Lipschitz continuous vector fields. In particular, this method retrieves the classical nonsmooth Newton method to solve a system of nonsmooth equations. The well-definedness of the sequence generated by the method is ensured and the convergence analysis of the method is made under local and semi-local assumptions.