In this work, a new full-wave strategy, with very low iteration times, is proposed for the optimization and design of antenna arrays fed by substrate integrated waveguides. The device is decomposed into fixed and modifiable (to be optimized) sections, whose generalized scattering matrices are precomputed. The response of the array is calculated by considering firstly only the interactions between the fixed sections, which are then coupled to the modifiable ones in each iteration of an optimization process. To validate the proposed strategy, a transversal 16-slot antenna array, placed on the top plate of a substrate integrated waveguide, has been designed. A speed-up factor of over 2000 times, compared to general purpose commercial software, has been obtained in this optimization process. The final design presents a 1.05 GHz bandwidth under - 10 dB in terms of |S11|, a maximum realized gain of 17.5 dBi at 17 GHz, and a 99.95% maximum efficiency (without dielectric and conductor losses). Index Terms—Addition theorems, cylindrical modes, spherical modes, optimization process, antenna array, substrate integrated waveguide (SIW).