A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming

TL;DR

This paper introduces a semi-smooth Newton method for a specific piecewise linear system, demonstrating linear convergence and effectiveness in solving large-scale positively constrained convex quadratic programming problems.

Contribution

It develops a semi-smooth Newton approach for a special piecewise linear system and applies it successfully to large-scale convex quadratic programming with positive constraints.

Findings

Method converges linearly under mild assumptions

Sequence generated by the method is bounded and has identifiable accumulation points

Achieves accurate solutions in few iterations for large-scale problems

Abstract

In this paper a special piecewise linear system is studied. It is shown that, under a mild assumption, the semi-smooth Newton method applied to this system is well defined and the method generates a sequence that converges linearly to a solution. Besides, we also show that the generated sequence is bounded, for any starting point, and a formula for any accumulation point of this sequence is presented. As an application, we study the convex quadratic programming problem under positive constraints. The numerical results suggest that the semi-smooth Newton method achieves accurate solutions to large scale problems in few iterations.