In this work a synthesis technique for designing sparse phased arrays of large size is presented. The procedure is based on an iterative convex optimization algorithm, also known as compressive sensing, which is straightforward to implement and efficient in terms of computational cost. It is capable of producing a layout with a significant reduction of active elements for predefined performance. Due to its effectiveness, it can also be employed for the optimization of large arrays. The capabilities and versatility of the algorithm are illustrated by different case studies. In particular, the possibility to reduce by 45% the number of elements in a large array is shown in details.