Semismooth Newton methods for the cone spectrum of linear transformations relative to Lorentz cones

We propose two semismooth Newton methods for seeking the eigenvalues of a linear transformation relative to Lorentz cones, by the natural equation reformulation and the normal equation reformulation, respectively, and establish their local quadratic convergence results under suitable conditions. The convergence analysis shows that the method based on the natural equation formulation is not influenced by the asymmetry of linear transformations, but the one based on the normal equation formulation suffers from this easily. Numerical experiments indicate that the method based on the normal equation formulation is very effective for the Lorentz eigenvalue problem of Z-transformations, and the method based on the natural equation formulation is very promising for those Lorentz eigenvalue problems of general linear asymmetric transformations with dimension less than 200, if one aims at finding at least one Lorentz eigenvalue.