  We know that compressive sensing can establish stable sparse recovery results
from highly undersampled data under a restricted isometry property condition.
In reality, however, numerous problems are coherent, and vast majority
conventional methods might work not so well. Recently, it was shown that using
the difference between $\ell_1$- and $\ell_2$-norm as a regularization always
has superior performance. In this paper, we propose an adaptive
$\ell_p$-$\ell_{1-2}$ model where the $\ell_p$-norm with $p\geq 1$ measures the
data fidelity and the $\ell_{1-2}$-term measures the sparsity. This proposed
model has the ability to deal with different types of noises and extract the
sparse property even under high coherent condition. We use a proximal
majorization-minimization technique to handle the nonconvex regularization term
and then employ a semismooth Newton method to solve the corresponding convex
relaxation subproblem. We prove that the sequence generated by the semismooth
Newton method admits fast local convergence rate to the subproblem under some
technical assumptions. Finally, we do some numerical experiments to demonstrate
the superiority of the proposed model and the progressiveness of the proposed
algorithm.
