  In the paper, we proposed the Dantzig selector based on the $\ell_{1}-\alpha
\ell_{2}$~$(0< \alpha \leq1)$ minimization for the signal recovery. In the
Dantzig selector, the constraint $\|{\bf A}^{\top}({\bf b}-{\bf A}{\bf
x})\|_\infty \leq \eta$ for some small constant $\eta>0$ means the columns of
${\bf A}$ has very weakly correlated with the error vector ${\bf e}={\bf A}{\bf
x}-{\bf b}$. First, recovery guarantees based on the restricted isometry
property (RIP) are established for signals. Next, we propose the effective
algorithm to solve the proposed Dantzig selector. Last, we illustrate the
proposed model and algorithm by extensive numerical experiments for the
recovery of signals in the cases of Gaussian, impulsive and uniform noise. And
the performance of the proposed Dantzig selector is better than that of the
existing methods.
