On the implementation of ADMM with dynamically configurable parameter for the separable \(\ell _{1}/\ell _{2}\) minimization

In this paper, we propose a novel variant of the alternating direction method of multipliers (ADMM) approach for solving minimization of the rate of \(\ell _{1}\) ${\ell }_1$ and \(\ell _{2}\) ${\ell }_2$ norms for sparse recovery. We first transform the quotient of \(\ell _{1}\) ${\ell }_1$ and \(\ell _{2}\) ${\ell }_2$ norms into a new function of the separable variables using the least squares minimum norm solution of the linear system of equations. Subsequently, we employ the augmented Lagrangian function to formulate the corresponding ADMM method with a dynamically adjustable parameter. Additionally, each of its subproblems possesses a unique global minimum. Finally, we present some numerical experiments to demonstrate our results.