Iterative reweighted minimization methods for l_p regularized unconstrained nonlinear programming

摘要

In this paper we study general l _p 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 l _p minimization problems. We extend some existing iterative reweighted l_1 (IRL_1) and l_2 (IRL_2) 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 thesemethods. In addition, we propose a novel Lipschitz continuous ?-approximation to ||x||_p~p. Using this result, we develop new IRL_1 methods for the l _p minimization problems and show that any accumulation point of the sequence generated by thesemethods 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 ? be dynamically updated and approach zero. Our computational results demonstrate that the new IRL_1 method and the new variants generally outperform the existing IRL_1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395-407, 2009).