A modified exact penalty approach for general constrained $\ell_0$-sparse optimization problems

Abstract: We consider a general class of constrained optimization problems with an additional $\ell_0$- sparsity term in the objective function. Based on a recent reformulation of this difficult $\ell_0$-term, we consider a nonsmooth penalty approach which differs from the authors previous work by the fact that it can be directly applied to problems which do not necessarily contain nonnegativity constraints. This avoids a splitting of free variables into their positive and negative parts, reduces the dimension and fully exploits the one-to-one correspondence between local and global minima of the given $\ell_0$-sparse optimization problem and its reformulation. The penalty approach is shown to be exact in terms of minima and stationary points. Since the penalty function is (mildly) nonsmooth, we also present practical techniques for the solution of the subproblems arising within the penalty formulation. Finally, the results of an extensive numerical testing are provided.