Sparse optimization over polyhedral sets based on weighted ℓ-norm minimization

Abstract

Finding the sparsest solutions over a polyhedral set is a challenging problem that has received a great deal of attention in the research community. This problem is typically NP-hard owing to the non-convexity and non-continuity of the ℓ 0 -norm. In this study, it is demonstrated that any solution of ℓ 0 -minimization over a polyhedral set must be one of the ‘extreme efficient solutions’ (EESs) of a special multiobjective optimization problem. Moreover, a new weighted ℓ 1 -norm minimization method using the Goldman–Tucker Theorem for linear optimization problems is established. Some essential features of this algorithm are examined, and numerical tests are performed to illustrate the efficiency of the proposed strategy. The numerical performance of the proposed method is compared with conventional reweighted ℓ 1 -algorithms and several compressed sensing solvers. Additionally, the method is applied to address the compressed sensing problem in magnetic resonance imaging (MRI) reconstruction.