An Exact Penalty Approach for General $ \ell_0 $-Sparse Optimization Problems
Abstract
We consider the general nonlinear optimization problem where the objective function has an additional term defined by the -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 -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.
Cite
@article{arxiv.2312.15706,
title = {An Exact Penalty Approach for General $ \ell_0 $-Sparse Optimization Problems},
author = {Christian Kanzow and Felix Weiß},
journal= {arXiv preprint arXiv:2312.15706},
year = {2023}
}