  In this paper, we consider a class of non-convex and non-smooth sparse
optimization problems, which encompass most existing nonconvex
sparsity-inducing terms. We show the second-order optimality conditions only
depend on the nonzeros of the stationary points. We propose two damped
iterative reweighted algorithms including the iteratively reweighted $\ell_1$
algorithm (DIRL$_1$) and the iteratively reweighted $\ell_2$ (DIRL$_2$)
algorithm, to solve these problems. For DIRL$_1$, we show the reweighted
$\ell_1$ subproblem has support identification property so that DIRL$_1$
locally reverts to a gradient descent algorithm around a stationary point. For
DIRL$_2$, we show the solution map of the reweighted $\ell_2$ subproblem is
differentiable and Lipschitz continuous everywhere. Therefore, the map of
DIRL$_1$ and DIRL$_2$ and their inverse are Lipschitz continuous, and the
strict saddle points are their unstable fixed points. By applying the stable
manifold theorem, these algorithms are shown to converge only to local
minimizers with randomly initialization when the strictly saddle point property
is assumed.
