An Exact Penalty Approach for General ${\mathrm{\ell }}_0$-Sparse Optimization Problems

We consider the general nonlinear optimization problem where the objective function has an additional term defined by the ${\mathrm{\ell }}_0$-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 ${\mathrm{\ell }}_0$-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.