Computational Experiments with Abs Algorithms for KKT Linear Systems

TL;DR

This work addresses solving KKT linear systems efficiently and accurately by evaluating ABS algorithms—specifically Huang, modified Huang, and implicit LU variants—and comparing them with LAPACK-based solvers. The KKT system is structured as $\begin{bmatrix} B & A^T\\ A & 0 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}=\begin{bmatrix} b\\ c \end{bmatrix}$ with $A \in \mathbb{R}^{m\times n}$, $B \in \mathbb{R}^{n\times n}$ symmetric, and $m\le n$. The main findings show that implicit LU variants provide the best total time when $m\sim n$, while a direct complete-KKT solve via Bunch-Parlett is faster for $m\ll n$; the modified Huang method, though slower overall, yields superior accuracy in highly ill-conditioned cases and can detect numerical rank. Range-space is the least accurate, and LAPACK-based approaches are competitive except under severe ill-conditioning, collectively guiding method choice for ABS-based KKT solvers.

Abstract

The results of computational experiments with ABS algorithms for KKT linear systems are reported.