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

Zhaosong Lu (zhaosong***at***sfu.ca)

Abstract: 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.

Keywords: $l_p$ minimization, iterative reweighted $l_1$ minimization, iterative reweighted $l_2$ minimization

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Unconstrained Optimization )

Citation: Manuscript, Department of Mathematics, Simon Fraser University, September 2012.

Download: [PDF]

Entry Submitted: 09/28/2012
Entry Accepted: 09/28/2012
Entry Last Modified: 10/06/2012

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