A Smoothing Iterative Method For L₁-L₀ Minimization

Sparse minimization has attracted much attention in compressed sense,signal reconstruc-tion,image restoration and economics so on.Sparse optimization problem are solved by finding an optimal solution of a minimization problem.The minimization problem studied in this paper is the least absolute deviation with the nonconvex lo penalty.Because the objective function of the least absolute deviation with the nonconvex lo penalty problem is nonconvex nonsmooth,we use the corresponding Huber penalty function to smooth the least absolute deviation(LAD)with the nonconvex lo penalty.We introduce the properties of corresponding quantile Huber penalty,and we obtain a e-approximation of the norm of lo.In addition,We establish the first lower bound and the second order bound for the absolute value of nonzero entries in every local optimal solution of the least absolute deviation with the nonconvex lo penalty problem.Finally we use the new smoothing iterative reweighted l1 minimization algorithm to solve approximate problems,and the algorithm is global convergence analysis is carried out.Our computational results show the effectiveness of the algorithm.This paper is organized as follow.In Section l,we introduce the background of research,the status of research,the main content and the frame structure of the whole paper.In Section 2,we introduce the original and the approximate problem of sparse recovery problem,and some preliminary knowledge which is prepared for the subsequent proofs of the algorithm.In Section 3,the least absolute deviation is smoothed by smoothing technique and the parameters are intro-duced to obtain the approximate model,which is solved by a new smoothing iterative algorithm.The convergence of the algorithm is proved.In Section 4,computational experiments are given to demonstrate the effectiveness of the algorithm.