Partial differential equation based inverse problem such as image reconstruction in electrical impedance tomography (EIT) often lead to non-convex optimization problems. These problems are also non-smooth when total variation (TV) type regularization is employed. To improve upon current optimization methods, such as (semi-smooth) Newton’s method and non-linear primal dual proximal splitting, we propose a new alternative: a relaxed inexact proximal Gauss-Newton method. We discuss the implementation of this method and investigate its convergence properties both theoretically and numerically with experimental EIT studies. This is a joint work with Petri Kuusela, Aku Seppänen and Tuomo Valkonen.