Research On Smoothing Algorithm Based On Sparse Optimization Lp Regularization

In recent years,the sparse optimization problem has attracted extensive attention in many fields such as scientific research and engineering practice.After Compressive sensing(CS)theory is proposed,sparse optimization has been widely applied in the fields of signal recovery,image processing and statistical inference.Theref'ore,the study of'sparse optimization problems has important theoretical and practical significance.CS theory is mainly divided into three parts:sparse representation,coding measurement and signal reconstruction.The core idea is to compress and sample the sparse signal,and then reconstruct the original signal through observation vector and algorithm.The information sampling theory has a great change due to the birth of CS?which has a good application prospect.The content of this paper is mainly based on the sparse optimization problem of lp(0<p<1)regularization.The sparse optimization problem of lp regularization is a kind of nonconvex nonsmooth non-Lipschitz continuous optimization problem.The model's objective function consists of the smooth term f(x)(usually 1/2||Ax—b||22)and nonconvex nonsmooth non-Lipschitz continuous regular term ||x||ppconsists of two parts.Solving the sparse optimization problem based on lp regularization needs to overcome the f'ollowing three problems,namely:(1)Since the minimization model is non-convex non-smooth non-Lipschitz continuous,it is not easy to establish its optimality condition.(2)Due to the large scale of'problems in real lif'e and engineering applications,the robust-ness,timeliness and ability to solve the algorithm are required to be high;(3)Existing algorithms have good performance in relatively easy sparse signal recovery problems.However,for difficult problems with high sparsity,carrying noise or low sampling rate,the recovery ability is limited.In order to solve the above problem,this paper performs two different smoothing processes for the regular item ||x||pp.The first smoothing process utilizes the idea of it-erative re-weighting algorithm.A fast iterative re-weighting algorithm based on sparse optimization lp regularization is proposed.The gradient equation of the smoothed ob?jective function is localized at each iteration.The linearization process greatly speeds up the running speed of'the algorithm,and proves the convergence of the algorithm and analyzed the computational complexity of the algorithm.The second smoothing process proposes a smoothing quasi-Newton algorithm based on sparse optimization lp regular-ization,a smoothing conjugate gradient algorithm and a smoothing modified Newton algorithm by constructing a smooth approximation function of the objective function.The convergence proves the convergence of the first two algorithms and analyzed their computational complexity separately.In the smoothing modified Newton algorithm,the Hessian matrix is modified by the characteristics of the model,and the p approaches to 0.Finally,the computational complexity of'the algorithm is analyzed.The adaptive adjustment of the smoothing parameters and the regularization parameters makes the above algorithms have wide adaptability and robustness.The effectiveness of the above algorithms is verified by a large number of numerical experiments.