Computation of second-order directional stationary points for group sparse optimization

We consider a nonconvex and nonsmooth group sparse optimization problem where the penalty function is the sum of compositions of a folded concave function and the vector norm for each group variable. We show that under some mild conditions a first-order directional stationary point is a strict local minimizer that fulfils the first-order growth condition, and a second-order directional stationary point is a strong local minimizer that fulfils the second-order growth condition. In order to compute second-order directional stationary points, we construct a twice continuously differentiable smoothing problem and show that any accumulation point of the sequence of second-order stationary points of the smoothing problem is a second-order directional stationary point of the original problem. We give numerical examples to illustrate how to compute a second-order directional stationary point by the smoothing method.