Complementarity Formulations of `0-norm Optimization Problems

In a number of application areas, it is desirable to obtain sparse solutions. Minimizing the number of nonzeroes of the solution (its `0-norm) is a difficult nonconvex optimization problem, and is often approximated by the convex problem of minimizing the `1-norm. In contrast, we consider exact formulations as mathematical programs with complementarity constraints and their reformulations as smooth nonlinear programs. We discuss properties of the various formulations and their connections to the original `0-minimization problem in terms of stationarity conditions, as well as local and global optimality. Numerical experiments using randomly generated problems show that standard nonlinear programming solvers, applied to the smooth but nonconvex equivalent reformulations, are often able to find sparser solutions than those obtained by the convex `1-approximation.