| Summary: | In this paper, we propose, analyze, and test an alternative method for solving the <inline-formula> <math display="inline"> <semantics> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>-norm regularization problem for recovering sparse signals and blurred images in compressive sensing. The method is motivated by the recent proposed nonlinear conjugate gradient method of Tang, Li and Cui [Journal of Inequalities and Applications, 2020(1), 27] designed based on the least-squares technique. The proposed method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and an <inline-formula> <math display="inline"> <semantics> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>-norm regularization term. The search directions generated by the proposed method are descent directions. In addition, under the monotonicity and Lipschitz continuity assumption, we establish the global convergence of the method. Preliminary numerical results are reported to show the efficiency of the proposed method in practical computation.
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