TY - JOUR
T1 - Equivalence Between Nonnegative Solutions to Partial Sparse and Weighted $l_1$-Norm Minimizations
AU - Tian , Xiuqin
AU - Dong , Zhengshan
AU - Zhu , Wenxing
JO - Annals of Applied Mathematics
VL - 4
SP - 380
EP - 395
PY - 2022
DA - 2022/06
SN - 32
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/aam/20650.html
KW - compressed sensing, sparse optimization, range space property, equivalent condition, $l_0$-norm minimization, weighted $l_1$-norm minimization.
AB - <p style="text-align: justify;">Based on the range space property (RSP), the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted $l_1$-norm minimization problem are studied in this paper. Different from
other conditions based on the spark property, the mutual coherence, the null
space property (NSP) and the restricted isometry property (RIP), the RSP-based conditions are easier to be verified. Moreover, the proposed conditions
guarantee not only the strong equivalence, but also the equivalence between
the two problems. First, according to the foundation of the strict complementarity theorem of linear programming, a sufficient and necessary condition,
satisfying the RSP of the sensing matrix and the full column rank property of
the corresponding sub-matrix, is presented for the unique nonnegative solution
to the weighted $l_1$-norm minimization problem. Then, based on this condition,
the equivalence conditions between the two problems are proposed. Finally,
this paper shows that the matrix with the RSP of order $k$ can guarantee the
strong equivalence of the two problems.</p>