Smoothing inertial neurodynamic approach for sparse signal reconstruction via L_p-norm minimization

In this paper, we propose a smoothing inertial neurodynamic approach (SINA) which is used to deal with L_p-norm minimization problem to reconstruct sparse signals. Note that the considered optimization problem is nonsmooth, nonconvex and non-Lipschitz. First, the problem is transformed into a smooth optimization problem based on smoothing approximation method, and the Lipschitz property of gradient of the smooth objective function is discussed. Then, SINA based on Karush-Kuhn-Tucker (KKT) condition, smoothing approximation and inertial dynamical approach, is designed to handle smooth optimization problem. The existence, uniqueness, global convergence and optimality of the solution of the SINA are discussed by the Cauchy-Lipschitz-Picard theorem, energy function and KKT condition. In addition, for p=1, the SINA has a mean sublinear convergence rate O1∕t under some mild conditions. Finally, some numerical examples on sparse signal reconstruction and image restoration are given to illustrate the theoretical results and the efficiency of SINA.