## 2 Description of the Method  
Let us consider an Uniform Linear Array (ULA) of \(N\) isotropic elements radiating in the free space, with inter-element distance \(d=\lambda/2\), where \(\lambda\) being the free space wavelength, see Figure 1. The goal is to identify the presence of faulty elements using a smaller number of far-field measurements. Following the approach outlined in [10] we suppose that the excitations of the failure-free array are available, as well as the corresponding far-field data in \(M\) measurement points. The failure-free array (i.e., the gold array) will be denoted as reference array. Let \(\mathbf{y}^{r}\in\mathbb{C}^{M}\) be the vector collecting the far-field of the failure-free array (wherein the superscript \(r\) stands for reference), and \(\mathbf{x}^{r}\in\mathbb{C}^{N}\) the vector of its excitations. The field radiated by the Antenna Under Test (AUT) with faulty elements is collected at the same spatial positions in the vector \(\mathbf{y}^{att}\in\mathbb{C}^{M}\), while the vector of its excitations will be denoted as \(\mathbf{x}^{att}\in\mathbb{C}^{N}\). Now, let us consider the linear system  
\[\mathbf{y}=\mathbf{A}\mathbf{x} \tag{1}\]  
wherein \(\mathbf{x}=\mathbf{x}^{r}-\mathbf{x}^{att}\) and \(\mathbf{y}=\mathbf{y}^{r}-\mathbf{y}^{att}\) are innovation vectors and \(\mathbf{A}\in\mathbb{C}^{M\times N}\) is the radiation matrix relating the excitations to the far-field data. Figure 1: Geometry of the Array Under Test. Since the number of faulty elements \(S\) is much smaller than the total number of array elements \(N\) (as usually happens), we have an equivalent problem involving a highly sparse array, in which only the faulty elements of the original array radiate. In other words, to locate the failed array elements is transformed into reconstruct a sparse vector accurately with high probability. Once recovered, the positions and numbers of non-zero entities represent the locations and amounts of the failed elements, respectively. A more general scenario is that the far-field measurement data are contaminated by a complex white Gaussian noise \(\mathbf{e}\) with zero mean and variance \(\sigma\)  
\[\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{e} \tag{2}\]  
This under-determined system is usually solved by the following optimization strategy  
\[\text{arg}\min||\mathbf{x}||_{p}\ \ \text{s.t.}\ \ \|\mathbf{y}-\mathbf{A} \mathbf{x}\|_{2}\leq\epsilon \tag{3}\]  
wherein \(||\mathbf{x}||_{p}\) is an objective function that promoting the sparsity, and the subscript \(p\) stands for \(\ell_{p}\) norm. \(\epsilon\) is related to the uncertainty level affecting the data. In the open literature the most widely adopted sparse promoting function is the \(\ell_{1}\) norm of the vector \(\mathbf{x}\)[23; 24]. Under certain conditions it is equivalent to the so called \(\ell_{0}\) norm minimization [25]. These conditions involve the number of measurements, that must be large enough to assure a accurate reconstruction of the unknown vectors. An interesting question regards the possibility of a further reduction of the number of measurements using a different norm as sparse promoting function. In this paper, instead of using \(\ell_{1}\) norm we focus our attention on a non-convex norm of vector \(\mathbf{x}=(x_{1},x_{2},...,x_{N})\), defined as  
\[\|\mathbf{x}\|_{p}=\left(\sum_{i=1}^{N}|x_{i}|^{p}\right)^{1/p} \tag{4}\]  
In this case, the minimization problem (3) is not convex. However, non-convex minimization arises in many different branches of applied physics, and a large number of local minimizer have been proposed. All the algorithms explore the search space picking up elements of the space according to some elaborate-designed rules. In our study we focus our attention on MCCA (Minimizing a Concave Function via a Convex Function Approximation) algorithm proposed by [26], as a local minimizer. The algorithm locally replacing the original concave objective function by a convex function.