The emergence and improvement of Additive Manufacturing technologies allow the fabrication of complex shapes so far inconceivable. However, to produce those intricate geometries, support structures are required. Besides wasting unnecessary material, these structures are consuming valuable production and post-processing times. This paper proposes a new framework to optimize the geometry and topology of inner and outer support structures. Starting from a uniform lattice structure filling both the inner and outer areas to be supported, the approach removes a maximum number of beams so as to minimize the volume of the support. The most suited geometry for the initial lattice structure is defined at the beginning considering the possibilities of the manufacturing technologies. Then, the pruning of the structure is performed through a genetic algorithm, for which the control parameters values have been tuned through a design of experiments. The proposed approach is validated on several test cases of various geometries, containing both inner and outer areas to be supported. The generated support structures are compared to the ones obtained by several state-of-the-art support structure strategies and are proved to have lower material consumption.