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 ${\ell }_2$ 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.

我们考虑一个非凸且非光滑的群稀疏优化问题，其中罚函数是折叠凹函数和 ${\ell }_2$每个组变量的向量范数。我们表明，在某些温和条件下，一阶有向平稳点是满足一阶增长条件的严格局部最小化子，而二阶有向平稳点是满足二阶增长条件的强大局部最小化子。为了计算二阶有向平稳点，我们构造了一个连续两次可微化的平滑问题，并证明了该平滑问题的二阶有平稳点序列的任何累积点都是原始问题的二阶有向平稳点。我们通过数值示例来说明如何通过平滑方法计算二阶有向固定点。