  We present an iterative support shrinking algorithm for $\ell_{p}$-$\ell_{q}$
minimization~($0 <p < 1 \leq q < \infty $). This algorithm guarantees the
nonexpensiveness of the signal support set and can be easily implemented after
being proximally linearized. The subproblem can be very efficiently solved due
to its convexity and reducing size along iteration. We prove that the iterates
of the algorithm globally converge to a stationary point of the
$\ell_{p}$-$\ell_{q}$ objective function. In addition, we show a lower bound
theory for the iteration sequence, which is more practical than the lower bound
results for local minimizers in the literature.
