Methods for constrained nonlinear systems: inexact Newton-like conditional gradient and Levenberg-Marquardt with inexact projections

Abstract

In this work, we propose and analyze some methods to solve constrained nonlinear systems of equations. First, we present a method that combines the inexact Newton-like method with a specialized version of the conditional gradient method (also known as Frank-Wolfe method). Using a majorant condition, which allows us to prove in a unified way convergence results for some classes functions, the local convergence of the proposed method as well as results on its rates are established. Second, we present a global version of the previous method by means of a derivative-free and nonmonotone line search strategy. Under appropriate conditions the global convergence of the proposed method is proved. Third, based on the well-known Levenberg-Marquardt method, we also propose a Levenberg-Marquardt method with inexact projections which combines the unconstrained Levenberg-Marquardt method with a notion of inexact projetions. In this case, the local convergence of the proposed method is proved using the error bound condition that is weaker than the standard condition full-rank of the Jacobian. Moreover, we also present a global version of the latter method by means of a nonmonotone line search technique. Finally, numerical experiments are also reported to illustrate the performances of the proposed schemes.