Iterative reweighted methods for ℓ1−ℓp minimization

Abstract

In this paper, we focus on the ℓ1−ℓpℓ1−ℓp minimization problem with 0<p<10<p<1, which is challenging due to the ℓpℓp norm being non-Lipschizian. In theory, we derive computable lower bounds for nonzero entries of the generalized first-order stationary points of ℓ1−ℓpℓ1−ℓpminimization, and hence of its local minimizers. In algorithms, based on three locally Lipschitz continuous ϵϵ-approximation to ℓpℓp norm, we design several iterative reweighted ℓ1ℓ1 and ℓ2ℓ2methods to solve those approximation problems. Furthermore, we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary point of ℓ1−ℓpℓ1−ℓp minimization. This result, in particular, applies to the iterative reweighted ℓ1ℓ1methods based on the new Lipschitz continuous ϵϵ-approximation introduced by Lu (Math Program 147(1–2):277–307, 2014), provided that the approximation parameter ϵϵ is below a threshold value. Numerical results are also reported to demonstrate the efficiency of the proposed methods.

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ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

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