Second-order dynamical system for solving $\ell _{1}-\ell _{2}$ minimization problem and its applications

\(\ell _{1}-\ell _{2}\) minimization problem finds extensive applications across various domains, including signal processing, face recognition, and image restoration. This study begins by reformulating \(\ell _{1}-\ell _{2}\) minimization model into an equivalent fixed-point formulation via a projection-based residual method. Based on this, a new second-order continuous-time projection (2-CTP) algorithm is constructed by leveraging the theory of second-order dynamical systems. Subsequently, the existence and uniqueness of the solution to the proposed algorithm are established, and the convergence of its trajectories is further analyzed. To illustrate the effectiveness and performance of the developed method, numerical simulations covering applications in signal processing and image restoration are provided in the end.