Electrical Impedance Tomography (EIT) seeks to recover the impedance distribution within a body using boundary data. More specifically, given the measured potentials, the model of the body - an elliptic partial differential equation - and the boundary conditions, this technique solves a non-linear inverse problem for the unknown impedance. In this work, an algorithm called Topology Optimization Method (TOM) is applied to EIT and compared to the Gauss-Newton Method (GNM). The Topology Optimization has solved some non-linear inverse problems and some of its procedures were not investigated for EIT, for instance, the use of Sequential Linear Programming. Assuming a pure resistive medium, the static resistivity distribution of a phantom was estimated using a 2-D finite element model. While the first method (GNM) essentially solves several algebraic systems, the second (TOM) solves several linear programming problems. Results using experimental data are shown and the quality of the images obtained, time and memory used are compared for both algorithms. We intend to use these methods, in future works, for the visualization of a human lung subjected to mechanical ventilation.