An Iterative Method for Solving Linear Inequalities

Abstract

This paper describes and analyzes a method for finding nontrivial solutions of the inequality $Ax \geq 0$, where $A$ is an $m \times n$ matrix of rank $n$. The method is based on the observation that a certain function $f$ has a unique minimum if and only if the inequality {\it fails to have} a nontrivial solution. Moreover, if there is a solution, an attempt to minimize $f$ will produce a sequence that will diverge in a direction that converges to a solution of the inequality. The technique can also be used to solve inhomogeneous inequalities and hence linear programming problems, although no claims are made about competitiveness with existing methods.