Newton’s method is an efficient and powerful procedure for solving nonlinear algebraic equations. In the
classical case, under suitable assumptions it will converge quadratically. One of its main drawback is its
sensitivity to initial iterates. In fact the basic method may diverge if the initial point is not chosen sufficiently
close to the solution.
Some extensions of Newton’s method to nonsmooth algebraic equations have been developed using
nonsmooth and variational analysis tools. There are many systems in engineering, economic, transportation
sciences, PDE’s and optimal control that can be formulated as systems of nonsmooth algebraic equations.
Recently, T. Hoheisel, C. Kanzow, B.S. Mordukhovich and H. Phan proved a convergence results without the
semismoothness assumption (only the continuity of the function is assumed) by using the graphical
derivatives. Assuming only the continuity of the function, is it possible to establish similar results to the
classical ones that can be obtained for a globalized version of the Newton methods for solving nonsmooth
The second part will be around the metric regularity in variational analysis.