This paper is closely related to [L. Kong et al., Pac. J. Optim. {\bf 18} (2022), no.~1, 213--232; MR4460526] and develops a new model consisting of a least absolute deviation loss function and an $L_{1/2}$-regularizer, which is a lower-order nonconvex, nonsmooth and non-Lipschitz optimization problem that is difficult to solve fast and efficiently. As a result, the problem is solved by an alternating direction method of multipliers (ADMM). Fortunately, all subproblems have closed-form solutions and convergence can be guaranteed. Numerical experiments show that the method has higher success probability when the number of measurements is relatively less and is robust to outliers and asymmetrical distribution noise, such as dense bounded noise and Laplace noise which yield advantageous statistical efficiency.