  We consider the general nonlinear optimization problem where the objective
function has an additional term defined by the $ \ell_0 $-quasi-norm in order
to promote sparsity of a solution. This problem is highly difficult due to its
nonconvexity and discontinuity. We generalize some recent work and present a
whole class of reformulations of this problem consisting of smooth nonlinear
programs. This reformulated problem is shown to be equivalent to the original $
\ell_0 $-sparse optimization problem both in terms of local and global minima.
The reformulation contains a complementarity constraint, and exploiting the
particular structure of this reformulated problem, we introduce several
problem-tailored constraint qualifications, first- and second-order optimality
conditions and develop an exact penalty-type method which is shown to work
extremely well on a whole bunch of different applications.
