Iterative reweighted minimization methods for lp 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 the 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 propose 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 ϵ-approximation to ‖x‖pp. Using this result, we develop new IRL1 methods for the lp minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter ϵ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that ϵ be dynamically updated and approach zero. Our computational results demonstrate that the new IRL1 method and the new variants generally outperform the existing IRL1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395–407, 2009).