## 1 Introduction  
Designing antennas for broadcast and telecommunication satellites presents significant challenges due to the stringent electrical and mechanical specifications required for space missions. Over several decades, reflector antennas with conformal surfaces have been employed to meet these rigorous standards [1]. However, in the past decade, reflectarray antennas have emerged as a compelling alternative, offering advantages such as simpler and more cost-effective manufacturing processes, a low profile, and independent polarization control [2]. A printed reflectarray antenna consists of multilayer substrates which host a planar array of resonant layouts, bounded below by a ground plane and illuminated by a feed antenna (usually a feed horn; see Figure 1) [2]. These resonant elements scatter the incident electric fields from the feed horn with a defined phase shift. Several techniques have been developed to generate shaped beams using reflectarrays [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. In [3], a comparison between three synthesis methods was carried out (an intersection approach with the MAP, a synthesis method based on the gradient technique method, and a synthesis method through beam addition). This comparison showed that the phase-only synthesis method, utilizing the intersection approach and MAP [4], proved to be highly effective for large reflectarrays [5]. In [6; 7; 8], the intersection approach and MAP using the Levenberg-Marquardt algorithm as a backward projector is shown. Although this approach showed significant improvements in accuracy, conventional alternating projections were implemented with non-convex sets [9], leading to a non-guaranteed convergence to an intersection point (if it exists). In [10], particle swarm optimization was carried out to optimize the phase shift distribution which generates shaped beams with a dual reflector system using penalty functions. In [11; 12], direct optimization of the dimensions of the layout was carried out to produce shaped beam reflectarrays (the phase shift distribution of the reflectarray elements was not considered). In [13], a direct optimization of the Phoenix cell was carried out through spherical mapping. All of these techniques could be extended to the case of aperiodic reflectarrays, since there are fast techniques for the computation of radiation patterns using non-uniform fast Fourier transform routines [14]. Although all of these methods provide good results in terms of convergence and accuracy, they are poorly scalable with respect to the size of a reflectarray. Therefore, it is preferable to use the intersection approach with alternating projections, because this approach has been proven to be highly effective for large reflectarrays [3; 5]. However, in [9], it was proven that the involved sets in the intersection approach are non-convex, and the MAP can converge to trap points. Therefore, another scheme for non-convex set treatments is preferred. Fortunately, there are several works where this scheme was studied. This new scheme is known as the alternating generalized projections (AGP) method. One method of AGP is known as the Douglas-Rachford method. Recent computational experiments have demonstrated the ability of Douglas-Rachford methods [15] to successfully solve a variety of significant non-convex optimization problems [16; 17; 18; 19; 20]. Other work can be found in [21] for multiplicative problems. There are other classic works, such as [22; 23]. Quite recent work can be found in [9], where the authors applied AGP methods for the phase-onlysynthesis of reflectarray antennas. In [9], the authors proposed AGP methods to escape from trap points. In our work, we will use the approach used in [9]. Next, we will briefly describe this approach. In the intersection approach, we consider two sets: the set of radiation patterns producible by the reflectarray (denoted as \(\mathcal{R}\)) and the set of radiation patterns which meet the mission's criteria (denoted as \(\mathcal{M}\)).