## 1 Introduction  
Array antennas are widely used in radar, remote sensing and mobile and satellite communications, etc. However, the deteriorated working condition, coupled with ever-growing of systematic complexity, inevitably increases the probability of defective array elements. Failed sensors impose negative impacts on the radiation performance, especially deteriorating the characteristic of beam-forming and lowering the accuracy of direction-of-arrival (DOA) estimation [1]. The cost associated with their life-cycle maintenance could exceed the corresponding original capital investment as the functionality of antennas and other equipments degrades as a result of age [2]. Therefore, before correcting the damaged array antennas, it is necessary to locate the numbers and positions of the broken elements via array diagnosis methods, which is of significant importance both on theoretical and practical values.  
Approaches for finding the impaired elements have been attracted considerable attentions of scholars and engineers worldwide, varying from academic community to industrial field. In specific, back propagation method [3], matrix method [4], neural networks [5], case-based reasoning [6], genetic algorithm [7], source reconstruction method [8] and mutual coupling technique [9] could be found in the open literature. Recently, the Compressed Sensing/Sparse Recovery (CS/SR) based approaches have also been proposed in the framework of array diagnosis [10; 11; 12; 13]. It is worth noting that, in general, accuracy and efficiency are two major aspects in developing diagnosis approaches. Therefore, improving the probability of success rate of diagnosis with an acceptable decrease of efficiency using either far-field or near-field measurements has always been the heated topic in the community of array diagnosis. For instance, re-weighted \(\ell_{1}\) norm minimization has been applied in both near-field case [14] and far-field case [15]. In [16] a fast diagnosis method of antenna arrays using a small number of far-field measurements with simulated and experimental data is addressed. Different from classical CS based paradigm, a reliable diagnosis method of both linear and planar arrays using Bayesian Compressive Sensing (BCS) are also proposed in [17; 18].  
The aim of this paper is to increase the probability of success rate of diagnosis with a smaller number of far-field measurements using a hybrid method of random perturbation and non-convex optimization, to overcome the issue of local minima associated to the intrinsic non-convex nature of \(\ell_{p}\) (\(0<p<1\)) norm minimization. The better performance of \(\ell_{p}\) (\(0<p<1\)) norm compared to \(\ell_{1}\) norm is not new in the open literature [19; 20; 21; 22]. However, application of non-convex norms in array diagnosis will bring unwanted impact on the performance of diagnosis unless the problem of solutions to be trapped in local minima is properly solved. In this article, lots of numerical experiments are carried out and the results have vividly confirmed that the proposed method overcomes this problem effectively, and also obtains a stronger promotion of sparsification of the solutions compared to the standard \(\ell_{1}\) norm minimization with an acceptable increase of calculated amount.