In this paper, we propose a novel algorithm that is based on quadratic-piecewise-linear approximations of DC functions to solve nonnegative sparsity-constrained optimization. A penalized DC (difference of two convex functions) formulation is proved to be equivalent to the original problem under a suitable penalty parameter. We employ quadratic-piecewise-linear approximations to the two parts of the DC objective function, resulting in a nonconvex subproblem. This is the key ingredient of our main algorithm. This nonconvex subproblem can be solved by a globally convergent alternating variable algorithm. Under some mild conditions, we prove that the proposed main algorithm for the penalized problem is globally convergent. Some preliminary numerical results on the sparse nonnegative least squares and logistic regression problems demonstrate the efficiency of our algorithm.

在本文中，我们提出了一种新的算法，该算法基于DC函数的二次分段线性逼近来解决非负稀疏约束优化问题。证明了在适当的惩罚参数下，惩罚DC（两个凸函数的差）公式等效于原始问题。我们对DC目标函数的两个部分采用了二次逐段线性逼近，从而产生了一个非凸子问题。这是我们主要算法的关键要素。该非凸子问题可以通过全局收敛的交替变量算法来解决。在某些温和条件下，我们证明了所提出的用于惩罚问题的主要算法是全局收敛的。