  This paper is devoted to studying the global and finite convergence of the
semi-smooth Newton method for solving a piecewise linear system that arises in
cone-constrained quadratic programming problems and absolute value equations.
We first provide a negative answer via a counterexample to a conjecture on the
global and finite convergence of the Newton iteration for symmetric and
positive definite matrices. Additionally, we discuss some surprising features
of the semi-smooth Newton iteration in low dimensions and its behavior in
higher dimensions. Moreover, we present two iterative schemes inspired by the
classical Jacobi and Gauss-Seidel methods for linear systems of equations for
finding a solution to the problem. We study sufficient conditions for the
convergence of both proposed procedures, which are also sufficient for the
existence and uniqueness of solutions to the problem. Lastly, we perform some
computational experiments designed to illustrate the behavior (in terms of CPU
time) of the proposed iterations versus the semi-smooth Newton method for dense
and sparse large-scale problems.
