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<p style='text-indent:20px;'>Finding sparse solutions to a linear system has many real-world applications. In this paper, we study a new hybrid of the <inline-formula><tex-math id="M3">\begin{document}$ l_p $\end{document}</tex-math></inline-formula> quasi-norm (<inline-formula><tex-math id="M4">\begin{document}$ 0 &lt;p&lt; 1 $\end{document}</tex-math></inline-formula>) and <inline-formula><tex-math id="M5">\begin{document}$ l_2 $\end{document}</tex-math></inline-formula> norm to approximate the <inline-formula><tex-math id="M6">\begin{document}$ l_0 $\end{document}</tex-math></inline-formula> norm and propose a new model for sparse optimization. The optimality conditions of the proposed model are carefully analyzed for constructing a partial linear approximation fixed-point algorithm. A convergence proof of the algorithm is provided. Computational experiments on image recovery and deblurring problems clearly confirm the superiority of the proposed model over several state-of-the-art models in terms of the signal-to-noise ratio and computational time.</p>