The optimization problem with sparsity arises in many areas of science and engineering such as compressed sensing, image processing, statistical learning and data sparse approximation. In this paper, we study the dual-density-based reweighted \(\ell _{1}\)-algorithms for a class of \(\ell _{0}\)-minimization models which can be used to model a wide range of practical problems. This class of algorithms is based on certain convex relaxations of the reformulation of the underlying \(\ell _{0}\)-minimization model. Such a reformulation is a special bilevel optimization problem which, in theory, is equivalent to the underlying \(\ell _{0}\)-minimization problem under the assumption of strict complementarity. Some basic properties of these algorithms are discussed, and numerical experiments have been carried out to demonstrate the efficiency of the proposed algorithms. Comparison of numerical performances of the proposed methods and the classic reweighted \(\ell _1\)-algorithms has also been made in this paper.

具有稀疏性的优化问题出现在科学和工程学的许多领域，例如压缩感测，图像处理，统计学习和数据稀疏近似。在本文中，我们研究了一类\（\ ell _ {0} \）-最小化模型的基于双密度的加权加权（\ ell_ {1} \）-算法，该模型可用于对宽范围进行建模一系列实际问题。此类算法基于底层\（\ ell _ {0} \）最小化模型的重新构造的某些凸松弛。这种重新构造是特殊的双层优化问题，从理论上讲，它等效于基础\（\ ell _ {0} \）严格互补假设下的最小化问题。讨论了这些算法的一些基本特性，并进行了数值实验以证明所提算法的有效性。本文还对所提方法和经典的加权（\ ell _1 \）算法的数值性能进行了比较。