Iterative Reweighted Minimization Methods for l_p Regularized Unconstrained Nonlinear Programming
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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 ϵ-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 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 is generally more stable than the existing IRL1 methods [21,18] in terms of objective function value and CPU time.
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