A Global and Quadratically Convergent Method for Linear $l_\infty $ Problems

Abstract

A new globally and quadratically convergent algorithm is proposed for the linear $l_\infty $ problem. This method works on the piecewise linear $l_\infty $, problem directly by generating descent directions—via a sequence of weighted least squares problems—and using a piecewise linear line search to ensure a decrease in the $l_\infty $ function at every step. It is proven that ultimately full Newton-like steps are taken where the Newton step is based on the complementary slackness condition holding at the solution. Numerical results suggest a very promising method; the number of iterations required to achieve high accuracy is relatively insensitive to problem size.