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The $\ell_p$ regularization problem with $0< p< 1$ has been widely studiedfor finding sparse solutions of linear inverse problems and gained successfulapplications in various mathematics and applied science fields. The proximalgradient algorithm is one of the most popular algorithms for solving the$\ell_p$ regularisation problem. In the present paper, we investigate thelinear convergence issue of one inexact descent method and two inexact proximalgradient algorithms (PGA). For this purpose, an optimality condition theorem isexplored to provide the equivalences among a local minimum, second-orderoptimality condition and second-order growth property of the $\ell_p$regularization problem. By virtue of the second-order optimality condition andsecond-order growth property, we establish the linear convergence properties ofthe inexact descent method and inexact PGAs under some simple assumptions. Bothlinear convergence to a local minimal value and linear convergence to a localminimum are provided. Finally, the linear convergence results of the inexactnumerical methods are extended to the infinite-dimensional Hilbert spaces.

Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems