We consider an \(\ell _0\)-minimization problem where \(f(x) + \gamma \Vert x\Vert _0\) is minimized over a polyhedral set and the \(\ell _0\)-norm regularizer implicitly emphasizes the sparsity of the solution. Such a setting captures a range of problems in image processing and statistical learning. Given the nonconvex and discontinuous nature of this norm, convex regularizers as substitutes are often employed and studied, but less is known about directly solving the \(\ell _0\)-minimization problem. Inspired by Feng et al. (Pac J Optim 14:273–305, 2018), we consider resolving an equivalent formulation of the \(\ell _0\)-minimization problem as a mathematical program with complementarity constraints (MPCC) and make the following contributions towards the characterization and computation of its KKT points: (i) First, we show that feasible points of this formulation satisfy the relatively weak Guignard constraint qualification. Furthermore, if f is convex, an equivalence is derived between first-order KKT points and local minimizers of the MPCC formulation. (ii) Next, we apply two alternating direction method of multiplier (ADMM) algorithms, named (ADMM\(_{\mathrm{cf}}^{\mu , \alpha , \rho }\)) and (ADMM\(_{\mathrm{cf}}\)), to exploit the special structure of the MPCC formulation. Both schemes feature tractable subproblems. Specifically, in spite of the overall nonconvexity, it is shown that the first update can be effectively reduced to a closed-form expression by recognizing a hidden convexity property while the second necessitates solving a tractable convex program. In (ADMM\(_{\mathrm{cf}}^{\mu , \alpha , \rho }\)), subsequential convergence to a perturbed KKT point under mild assumptions is proved. Preliminary numerical experiments suggest that the proposed tractable ADMM schemes are more scalable than their standard counterpart while (ADMM\(_{\mathrm{cf}}\)) compares well with its competitors in solving the \(\ell _0\)-minimization problem.

我们考虑一个\（\ ell _0 \）最小化问题，其中\（f（x）+ \ gamma \ Vert x \ Vert _0 \）在多面集上被最小化，并且\（\ ell _0 \）- norm正则化器隐式化强调解决方案的稀疏性。这样的设置捕获了图像处理和统计学习中的一系列问题。鉴于此规范的非凸性和不连续性，经常使用和研究凸正则化器作为替代品，但对于直接求解\（\ ell _0 \）最小化问题知之甚少。受到冯等人的启发。（Pac J Optim 14：273–305，2018年），我们考虑解决\（\ ell _0 \）的等价表述。-最小化问题作为具有互补约束（MPCC）的数学程序，并且对其KKT点的特征和计算做出了以下贡献：（i）首先，我们证明了该公式的可行点满足相对较弱的吉格纳约束条件。此外，如果f是凸的，则在一阶KKT点和MPCC公式的局部极小值之间得出等价关系。（ii）接下来，我们应用两种交替方向乘数（ADMM）算法，分别是（ADMM \（_ {\ mathrm {cf}} ^ {\ mu，\ alpha，\ rho} \））和（ADMM \（ _ {\ mathrm {cf}} \）），以开发MPCC配方的特殊结构。两种方案都具有易处理的子问题。具体而言，尽管整体上不具有凸性，但显示出通过识别隐藏的凸性，可以将第一更新有效地简化为闭合形式的表达式，而第二更新则需要求解可处理的凸程序。在（ADMM \（_ {\ mathrm {cf}} ^ {\ mu，\ alpha，\ rho} \）中，证明了在温和假设下随后收敛到扰动的KKT点。初步的数值实验表明，所提出的易处理ADMM方案比标准方案更具可扩展性，而（ADMM \（_ {\ mathrm {cf}} \））与它的竞争对手在解决\（\ ell _0 \）最小化方面比较好问题。