  Recently, finding the sparsest solution of an underdetermined linear system
has become an important request in many areas such as compressed sensing, image
processing, statistical learning, and data sparse approximation. In this paper,
we study some theoretical properties of the solutions to a general class of
$\ell_{0}$-minimization problems, which can be used to deal with many practical
applications. We establish some necessary conditions for a point being the
sparsest solution to this class of problems, and we also characterize the
conditions for the multiplicity of the sparsest solutions to the problem.
Finally, we discuss certain conditions for the boundedness of the solution set
of this class of problems.
