  We extend the perturbation theory of Vishik, Ljusternik and Lidskii for
eigenvalues of matrices, using methods of min-plus algebra. We show that the
asymptotics of the eigenvalues of a perturbed matrix is governed by certain
discrete optimisation problems, from which we derive new perturbation formulae,
extending the classical ones and solving cases which where singular in previous
approaches. Our results include general weak majorisation inequalities,
relating leading exponents of eigenvalues of perturbed matrices and min-plus
analogues of eigenvalues.
