On the Newton method for set-valued maps

Abstract

The Newton method is one of the most powerful tools used to solve systems of nonlinear equations. Its set-valued generalization, considered in this work, allows one to solve also nonlinear equations with geometric constraints and systems of inequalities in a unified manner. The emphasis is given to systems of linear inequalities. The study of the wellposedness of the algorithm and of its convergence is fulfilled in the framework of modern variational analysis.The authors are grateful to Oleg Burdakov, Boris Mordukhovich, and Vera Roshchina for their valuable comments. This research was supported by the Portuguese Foundation for Science and Technologies (FCT), the Portuguese Operational Programme for Competitiveness Factors (COMPETE), the Portuguese National Strategic Reference Framework (QREN), and the European Regional Development Fund (FEDER)