  The optimization problem of sparse and low-rank matrix recovery is
considered, which involves a least squares problem with a rank constraint and a
cardinality constraint. To overcome the challenges posed by these constraints,
an asymptotic difference-of-convex (ADC) method that employs a Moreau smoothing
approach and an exact penalty approach is proposed to transform this problem
into a DC programming format gradually. To solve the gained DC programming, by
making full use of its DC structure, an efficient inexact DC algorithm with
sieving strategy (siDCA) is introduced. The subproblem of siDCA is solved by an
efficient dual-based semismooth Newton method. The convergence of the solution
sequence generated by siDCA is proved. To illustrate the effectiveness of
ADC-siDCA, matrix recovery experiments on nonnegative and positive semidefinite
matrices. The numerical results are compared with those obtained using a
successive DC approximation minimization method and a penalty proximal
alternating linearized minimization approach. The outcome of the comparison
indicates that ADC-siDCA surpasses the other two methods in terms of efficiency
and recovery error. Additionally, numerical experiments on sparse phase
retrieval demonstrate that ADC-siDCA is a valuable tool for recovering sparse
and low-rank Hermitian matrices.
