We propose a novel, variational inversion methodology for the electrical impedance tomography (EIT) problem, where we seek electrical conductivity inside a bounded, simply connected domain , given simultaneous measurements of electric currents I and corresponding potentials Vat the boundary. Explicitly, we make use of natural, variational constraints on the space of admissible functions , to obtain efficient reconstruction methods which make best use of the data. We give a detailed analysis of the variational constraints; we propose a variety of reconstruction algorithms for the static problem in a simple continuum model. We discuss their advantages and disadvantages and we assess the performance of our algorithms through numerical simulations and comparisons with other, well established, numerical reconstruction methods.