In this article, a novel second-order projection neurodynamic optimization approach is proposed to solve \(\alpha l_1-\beta l_2\) sparsity regularization problem from a continuous-time perspective. We first reformulate the nonsmooth and nonconvex objective function to a new form by using the Huber function, the advantage is that the gradient of the objective function can satisfy the Lipschitz condition. Then, a necessary optimality condition for the problem is revealed and a dynamical system is presented to obtain the analytic solution of the problem. Furthermore, we prove that the strong global solution of this algorithm is unique by the Cauchy-Lipschitz-Picard theorem. Under some mild conditions, the convergence of the proposed algorithm is rigorously established. Finally, some simulation experiments of signal recovery demonstrate the effectiveness of the proposed algorithm, and we give an application of the proposed algorithm to gray-scale and color image reconstruction. Compared with other existing algorithms, the proposed projection neurodynamic optimization approach is significantly competitive.
