## 2 Description of the Method
Given a point \(t_{0}\) on the concave function, a quadratic convex function \(g(t)\) using the second-order Taylor series expansion around the point \(t_{0}\) with the sign of the second order term reversed is built. The position of minimum of this quadratic (convex) function, let \(t_{1}\) be is used to build a new quadratic function. In [26] it has shown that the sequences of minima of these quadratic functions converge to a fixed point that reduces the original concave function. A pseudo-code of this algorithm is also reported in the paper. It is also worth noting that there is no magic algorithm which is able to efficiently solve any minimization problems, as stated by the famous no free lunch theorem [27]. For \(\ell_{p}\) (\(0<p<1\)) norm minimization, the major drawback is that the solution could be trapped in local minima. The relevant consequence is that the use of local minimizer does not assure correct estimation of the minimization problem, which results in an adverse impact on the diagnosis performance. Therefore, the minimization algorithm must be tailored to the problem. In order to overcome the defect of MCCA, a hybrid method of random search-convex local minimizer is proposed. The use of hybrid algorithms is quite common in various minimization problems and consists of using a local minimizer whose starting point is the randomly selected value of the function. In our specific minimization problem, it is possible to restrict the search space to a few-dimensional space. In fact, given any solution of the problem \(\mathbf{A}\mathbf{x}=\mathbf{y}\), the sparsest solution belongs to the null space of matrix \(\mathbf{A}\) shifted by the solution. Consequently, a possible strategy is to find a solution of the minimization problem and then perform a random search in the null space of that matrix. This suggests the following strategy. (1) find the least squares solution of the problem \(\mathbf{A}\mathbf{x}=\mathbf{y}\), which is given by the pseudo-inverse of matrix \(\mathbf{A}\). Let \(\mathbf{s}_{0}\) as this initial starting point, that is associated to the \(\ell_{2}\) norm minimization [28]. (2) use MCCA to generate an initial solution vector \(\mathbf{x}_{0}\). (3) a random vector \(\mathbf{n}_{0}\) is introduced to perturb this initial solution vector \(\mathbf{x}_{0}\), where \(\mathbf{n}_{0}\) should belong to the null space of matrix \(\mathbf{A}\) and generate a new starting point \(\mathbf{s}_{1}\) by adding the random vector \(\mathbf{n}_{0}\) into the initial solution vector \(\mathbf{x}_{0}\), i.e., \(\mathbf{s}_{1}=\mathbf{x}_{0}+\mathbf{n}_{0}\). (4) use MCCA to get an updated solution vector \(\mathbf{x}_{1}\). (5) if the \(\ell_{p}\) (\(0<p<1\)) norm of solution vector \(\mathbf{x}_{1}\) is not larger than \(\mathbf{x}_{0}\), i.e., \(||\mathbf{x}_{1}||_{p}\leq||\mathbf{x}_{0}||_{p}\), accept \(\mathbf{x}_{1}\), otherwise accept \(\mathbf{x}_{0}\) as a new starting point \(\mathbf{s}_{2}\). (6) repeat step (1)-(5) until the stopping criterion \(Q\) is reached, where \(Q\) denotes the number of random starting points. The value of \(Q\) should be a trade-off between accuracy of diagnosis and the computational burden. In our approach the critical factor is the design of \(\mathbf{n}\) which is formulated as \(\mathbf{n}=\mathbf{Fu}\), where \(\mathbf{F}\) belongs to the null space of matrix \(\mathbf{A}\) and \(\mathbf{u}\in\mathbb{R}^{N-r}\) represents a random noise vector, respectively. One feasible way of generating \(\mathbf{F}\) and \(\mathbf{u}\) is given in the following steps. Let \((\mathbf{U},\mathbf{V},\Sigma)\) the singular system of matrix \(\mathbf{A}\), where \(\mathbf{U}\) and \(\mathbf{V}\) are matrices whose columns are respectively the left and right singular vectors of matrix \(\mathbf{A}\). \(\Sigma\) is a diagonal matrix whose diagonal elements \(\sigma_{k}\) (\(k=1,...,r\)) are the singular values of the matrix, where \(r\) being its rank.