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We consider an $\ell_0$-minimization problem where $f(x) + \gamma \|x\|_0$ isminimized over a polyhedral set and the $\ell_0$-norm regularizer implicitlyemphasizes sparsity of the solution. Such a setting captures a range ofproblems in image processing and statistical learning. Given the nonconvex anddiscontinuous nature of this norm, convex regularizers are often employed assubstitutes. Therefore, far less is known about directly solving the$\ell_0$-minimization problem. Inspired by [19], we consider resolving anequivalent formulation of the $\ell_0$-minimization problem as a mathematicalprogram with complementarity constraints (MPCC) and make the followingcontributions towards the characterization and computation of its KKT points:(i) First, we show that feasible points of this formulation satisfy therelatively weak Guignard constraint qualification. Furthermore, under thesuitable convexity assumption on $f(x)$, an equivalence is derived betweenfirst-order KKT points and local minimizers of the MPCC formulation. (ii) Next,we apply two alternating direction method of multiplier (ADMM) algorithms toexploit special structure of the MPCC formulation: (ADMM$_{\rm cf}^{\mu,\alpha, \rho}$) and (ADMM$_{\rm cf}$). These two ADMM schemes both havetractable subproblems. Specifically, in spite of the overall nonconvexity, weshow that the first of the ADMM updates can be effectively reduced to aclosed-form expression by recognizing a hidden convexity property while thesecond necessitates solving a convex program. In (ADMM$_{\rm cf}^{\mu, \alpha,\rho}$), we prove subsequential convergence to a perturbed KKT point under mildassumptions. Our preliminary numerical experiments suggest that the tractableADMM schemes are more scalable than their standard counterpart and ADMM$_{\rmcf}$ compares well with its competitors to solve the $\ell_0$-minimizationproblem.

Tractable ADMM Schemes for Computing KKT Points and Local Minimizers for $\ell_0$-Minimization Problems