An iterative support shrinking algorithm for ${\mathrm{\ell }}_p$-${\mathrm{\ell }}_q$ minimization

We present an iterative support shrinking algorithm for ${\mathrm{\ell }}_p$-${\mathrm{\ell }}_q$ minimization~($0<p<1\le q<\mathrm{\infty }$). 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 ${\mathrm{\ell }}_p$-${\mathrm{\ell }}_q$ 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.