On the convergence of iterative schemes for solving a piecewise linear system of equations

TL;DR

This paper investigates the convergence properties of semi-smooth Newton and other iterative methods for solving piecewise linear systems, revealing limitations and proposing alternative schemes with convergence analysis and computational experiments.

Contribution

It provides a counterexample to a conjecture on convergence, introduces new iterative schemes inspired by classical methods, and analyzes their convergence and computational performance.

Findings

Counterexample disproves conjecture on Newton convergence for positive definite matrices.

Proposed iterative schemes have sufficient conditions for convergence and solution uniqueness.

Computational experiments compare efficiency of methods on large-scale problems.

Abstract

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.