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

Author(s)

Armijo, Nicolas F. ; Bello-Cruz, Yunier ; Haeser, Gabriel

• Document type: Journal article
• Source: Linear Algebra and its Applications; v. 665, p. 24-pg., 2023-02-22.

Abstract

This paper is devoted to studying the global and finite conver-gence of the semi-smooth Newton method for solving a piece -wise 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 con-jecture on the global and finite convergence of the Newton iteration for symmetric and positive definite matrices. Ad-ditionally, we discuss some surprising features of the semi -smooth Newton iteration in low dimensions and its behavior in higher dimensions. Secondly, 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 prob-lem. We study sufficient conditions for the convergence of both proposed procedures, which are also sufficient for the exis-tence and uniqueness of solutions to the problem. Lastly, we perform some computational experiments designed to illus-trate the behavior (in terms of CPU time) of the proposed iterations versus the semi-smooth Newton method for dense and sparse large-scale problems. Moreover, we included the numerical solution of a discretization of the Boussinesq PDE modeling a two-dimensional flow in a homogeneous phreatic aquifer.(c) 2023 Published by Elsevier Inc. (AU)

• FAPESP's process: 19/13096-2 - Complementarity measures in optimality conditions
• Grantee: Nicolas Esteban Fuentealba Armijo
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)
• FAPESP's process: 18/24293-0 - Computational methods in optimization
• Grantee: Sandra Augusta Santos
• Support Opportunities: Research Projects - Thematic Grants