Sign controlled solvers for the absolute value equation with an application to support vector machines

TL;DR

This paper introduces three solvers for the absolute value equation, extending their convergence ranges, and demonstrates their effectiveness on support vector machine problems reformulated as such equations.

Contribution

It presents new variants of solvers for the absolute value equation and extends their convergence analysis, applied to support vector machine problems.

Findings

Solvers successfully handle absolute value equations in SVM contexts.

Extended convergence ranges improve solver reliability.

Performance comparisons show practical effectiveness.

Abstract

Let $A$ be a real $n\times n$ matrix and $z,b\in \mathbb R^n$. The piecewise linear equation system $z-A\vert z\vert = b$ is called an absolute value equation. It is equivalent to the general linear complementarity problem, and thus NP hard in general. Concerning the latter problem, three solvers are presented: One direct, one semi-iterative and one discrete variant of damped Newton. Their previously proved ranges of correctness and convergence, respectively, are extended. Their performance is compared on instances of the XOR separation problem for support vector machines which can be reformulated as an absolute value equation.