## 3 Numerical Simulations  
As first example let us consider the same ULA with \(N=32\) radiating elements reported in [29], to which the reader can refer for more details on the simulation procedure. All the elements have unit nominal excitations. We suppose that the AUT is affected by \(S=3\) faulty elements having both amplitude and phase failures, randomly distributed among the \(N\) elements, and a number of far-field measurements \(M=10\) are considered. Each excitation of faulty elements can be modeled as \(x=(1-\delta_{A})e^{i\phi_{0}}\), where \(0<\delta_{A}<1\) and \(0<\delta_{\phi}<10^{\theta}\) representing amplitude and phase failures, respectively. The far-field data are corrupted by a 35 dB additive white Gaussian noise. As discussed in the above section, in this paper the non-linear nature of \(\ell_{p}\) (\(0<p<1\)) norm minimization is alleviated by a random search associated to a convex local minimizer. The random starting points explored in the procedure are selected randomly in the shifted null space \(\mathcal{N}(\mathbf{A})\) of matrix \(\mathbf{A}\). In order to investigate the effectiveness of the proposed method, in Figures 2-4 the Cumulative Distribution Function (CDF) of the Mean Square Error (MSE) of the retrieved solutionsobtained by proposed method of random search-convex local minimizer is plotted considering three representative non-convex norms, i.e., strong non-convex norm (\(p=0.1\)), moderate non-convex norm (\(p=0.5\)) and weak non-convex norm (\(p=0.9\)) with the number of random starting points \(Q=5,10\), respectively. In the same figure the CDF curve of \(\ell_{p}\) (\(0<p<1\)) norm minimization using MCCA and the standard \(\ell_{1}\) norm minimization using CVX convex minimization software package [30] are also plotted for sake of comparison. For each case, a number of 500 trials have been considered, randomly changing the positions of the broken elements. The plot shows that a value of \(Q\) equal or higher than 10 is able to overcome the strong non-convex feature of \(\ell_{0.1}\) norm and provide better performance compared to standard \(\ell_{1}\) norm minimization. While for the moderate non-convex norm, i.e., \(p=0.5\), a smaller number of random starting points, i.e., \(Q=5\) is enough to have satisfactory result. An analysis of the computational time obtained using a Lenovo desktop with dual quad-core 3.6 GHz Intel i7-4790 processors and 8 GB of memory is shown in Table 1. It should be pointed out that  
Figure 3: CDF of MSE of the retrieved excitations using proposed method in the case \(Q=5\) and the standard \(\ell_{1}\) norm minimization; solid line: \(\ell_{1}\) norm minimization; dashed line: proposed method. Figure 2: CDF of MSE of the retrieved excitations using \(\ell_{0.1}\), \(\ell_{0.5}\), \(\ell_{0.9}\) norm minimization and the standard \(\ell_{1}\) norm minimization; solid line: \(\ell_{1}\) norm minimization; dashed line: \(\ell_{p}\) (\(0<p<1\)) norm minimization. the \(\ell_{p}\) (\(0<p<1\)) norm minimization is equivalent to using just only one random starting point, i.e., in the case \(Q=1\) using our method, as display in the first three rows of second column in Table 1. As mentioned above, for moderate non-convex norm (\(p=0.5\)), a value of \(Q=5\) is enough to overcome its strong non-convex feature, and showing better performance compared to the standard \(\ell_{1}\) norm minimization. The cost associated with this improvement is displayed in Table 1 that in the case \(Q=5\), the average computational time of proposed method is just around four times longer than the classical \(\ell_{1}\) norm minimization. This ratio is around 10 times longer compared to the standard \(\ell_{1}\) norm minimization when considering strong non-convex norm (\(p=0.1\)), requiring \(Q=10\) to achieve satisfactory result. Therefore, from the perspective of accuracy and efficiency, the increasement of computational burden appears to be acceptable for array diagnosis.