Iterative Reweighted Minimization Methods for $l_p$ Regularized Unconstrained Nonlinear Programming

In this paper we study general $lp$ 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 $lp$ minimization problems. We extend some existing iterative reweighted $l1$ (IRL1) and $l2$ (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.