A semi-smooth Newton method for solving convex quadratic programming problem under simplicial cone constraint

TL;DR

This paper introduces a semi-smooth Newton method for efficiently solving convex quadratic programming problems with simplicial cone constraints, demonstrating linear convergence and practical effectiveness for large-scale problems.

Contribution

It establishes a novel semi-smooth Newton approach for these problems, proving convergence and providing numerical evidence of efficiency.

Findings

Method converges linearly under mild conditions

Sequence generated by the method is bounded for any starting point

Numerical results show high accuracy with few iterations

Abstract

In this paper the simplicial cone constrained convex quadratic programming problem is studied. The optimality conditions of this problem consist in a linear complementarity problem. This fact, under a suitable condition, leads to an equivalence between the simplicial cone constrained convex quadratic programming problem and the one of finding the unique solution of a nonsmooth system of equations. It is shown that a semi-smooth Newton method applied to this nonsmooth system of equations is always well defined and under a mild assumption on the simplicial cone the method generates a sequence that converges linearly to its 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. The presented numerical results suggest that this approach achieves accurate solutions to large problems in few iterations.