  In this paper we study general $l_p$ regularized unconstrained minimization
problems. In particular, we derive lower bounds for nonzero entries of first-
and second-order stationary points, and hence also of local minimizers of the
$l_p$ minimization problems. We extend some existing iterative reweighted $l_1$
(IRL1) and $l_2$ (IRL2) minimization methods to solve these problems and
proposed new variants for them in which each subproblem has a closed form
solution. Also, we provide a unified convergence analysis for these methods. In
addition, we propose a novel Lipschitz continuous $\epsilon$-approximation to
$\|x\|^p_p$. Using this result, we develop new IRL1 methods for the $l_p$
minimization problems and showed that any accumulation point of the sequence
generated by these methods is a first-order stationary point, provided that the
approximation parameter $\epsilon$ is below a computable threshold value. This
is a remarkable result since all existing iterative reweighted minimization
methods require that $\epsilon$ be dynamically updated and approach zero. Our
computational results demonstrate that the new IRL1 method is generally more
stable than the existing IRL1 methods [21,18] in terms of objective function
value and CPU time.
