Antenna systems are of great importance for communication, sensing, radar systems, and etc. For example, in wireless communications, multiple input multiple output antenna systems have been widely studied to meet the requirements of the fifth generation mobile communication (Alibakhshikenari, Virdee, Benetatos, et al., 2022), and some metamaterial-based antenna systems are attractive for further millimeter-wave (mm-wave) and terahertz (THz) wireless communications Alibakhshikenari, Virdee, Althuwayb, et al., 2021). In radar systems, phased arrays are widely used because of their pattern mainlobe shape and sidelobe level (SLL) control capability, as well as some other superior properties (Fenn et al., 2000;Kouemou, 2010). However, it is known that antenna arrays are usually used together with radomes which protect antenna arrays from the influence of outside environments in practice. The presence of the radome usually degrades the pattern performance of the antenna array (Frech et al., 2013;L. Liu & Nie, 2019). Thus, to suppress the radome effect, a number of techniques have been presented for optimizing radome's parameters such as its shape and thickness distribution (Hsu et al., 1993;Xu et al., 2017). However, the freedom of radome optimization is sometimes very limited since the radome should also meet other requirements such as its mechanical strength and electromagnetic scattering characteristics. Compared with the modification of a radome, optimizing element excitation amplitudes and phases by considering the antenna array-radome as a whole system would be a more convenient alternative to compensate for the radome effect if a phased antenna array is considered.In the past, many array pattern synthesis methods have been presented, but most of them deal only with the array factor synthesis, ignoring the mutual coupling (Ares-