This paper is devoted to the construction and analysis of a novel hybrid optimization model of the ${\ell }_0$ minimization and the ${\ell }_1$ minimization with a given support estimate. With the help of the Moreau envelop of the ${\ell }_1$ norm, we provide reasonable explanation for the claim that the capped-${\ell }_1$ penalty is one of the continuous relaxation to the ${\ell }_0$-norm penalty and thus develop the scale iteratively reweighed ${\ell }_1$-minimization (SIRL1) aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the ${\ell }_1$ minimization approach. To illustrate the theoretical results, some numerical experiments are presented to demonstrate the effectiveness and flexibility of the proposed SIRL1 algorithm.

本文致力于构建和分析一种新的混合优化模型 ${\ell }_0$ 最小化和 ${\ell }_1$使用给定的支持估计进行最小化。在 Moreau 包络的帮助下${\ell }_1$ 规范，我们为上限的说法提供了合理的解释 -${\ell }_1$ 惩罚是对 ${\ell }_0$- 规范惩罚，从而开发迭代重新加权的规模 ${\ell }_1$-minimization (SIRL1) 旨在实现快速重建和减少测量次数的要求 ${\ell }_1$最小化方法。为了说明理论结果，提出了一些数值实验来证明所提出的 SIRL1 算法的有效性和灵活性。