  We propose an exact iterative algorithm for minimization of a class of
continuous cell-wise linear convex functions on a hyperplane arrangement. Our
particular setup is motivated by evaluation of so-called rank estimators used
in robust regression, where every cell of the underlying arrangement
corresponds to a permutation of residuals (and we also show that the class of
function for which the method works is more general). The main obstacle in the
construction of the algorithm is how to find an improving direction while
standing in a point incident with exponentially many cells of the arrangement.
We overcome this difficulty using Birkhoff Theorem which allows us to express
the cone of improving directions in the exponential number of cells using a
linear system with quadratic number of variables only.
