  In this paper, we investigate a group sparse optimization problem via
$\ell_{p,q}$ regularization in three aspects: theory, algorithm and
application. In the theoretical aspect, by introducing a notion of group
restricted eigenvalue condition, we establish some oracle property and a global
recovery bound of order $O(\lambda^\frac{2}{2-q})$ for any point in a level set
of the $\ell_{p,q}$ regularization problem, and by virtue of modern variational
analysis techniques, we also provide a local analysis of recovery bound of
order $O(\lambda^2)$ for a path of local minima. In the algorithmic aspect, we
apply the well-known proximal gradient method to solve the $\ell_{p,q}$
regularization problems, either by analytically solving some specific
$\ell_{p,q}$ regularization subproblems, or by using the Newton method to solve
general $\ell_{p,q}$ regularization subproblems. In particular, we establish
the linear convergence rate of the proximal gradient method for solving the
$\ell_{1,q}$ regularization problem under some mild conditions. As a
consequence, the linear convergence rate of proximal gradient method for
solving the usual $\ell_{q}$ regularization problem ($0<q<1$) is obtained.
Finally in the aspect of application, we present some numerical results on both
the simulated data and the real data in gene transcriptional regulation.
