  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.
