Research On Accelerated Algorithms For Non-Lipschitz Sparse Optimization Models

This thesis mainly focuses on the algorithm design,convergence analysis and con-vergence rate of non-convex non-Lipschitz models in sparse optimization problems.Previous studies have shown that non-convex regularized models have advantages over convex regularized models in sparse optimization problems,for example,non-convex regularized models can get sparser solutions under the same observation data.How-ever,the algorithm design and convergence analysis of non-convex regularization mod-els are more difficult.In recent years,research on the KL property has made it feasible to establish the convergence and convergence rate of some non-convex optimization algorithms.In this paper,our contributions are mainly two works in Chapter 2 and Chapter 3In Chapter 2,we consider the p(0<p<1)regularization synthesis model for wavelet-based image restoration,which is a non-convex non-Lipschitz model.The Isoft algorithm designed for the p regularization model was proposed in paper[1].This algorithm is computationally efficient,but whether it converges or not has not been concluded yet.In this paper,we firstly reformulate the Isoft algorithm into the ISSAFL form.Then,we propose a new algorithm based on Nesterov’s extrapolation technique,i.e.,AISSAFL.Furthermore,a complete convergence analysis of AISSAFL is established by the KL property.We prove that the whole sequence generated by AIS-SAFL converges to a stationary point of the objective function.This convergence result contains the convergence analysis of the Isoft algorithm as a special case.We apply the AISSAFL algorithm to image deblurring and CT image reconstruction,numerical experiments demonstrate a good performance of this algorithm.In Chapter 3,we propose a new algorithm named as AIS~3for a general non-convex non-Lipschitz model on the group sparse optimization problem.The lower bound theory of the nonconvex non-Lipschitz model contains a group support set shrinking strategy,therefore,we use this strategy to overcome the non-Lipschitz property of this model,the group support set shrinking strategy can also reduce the scale of the prob-lem in every iteration.In numerical experiments,considering the finite word length of real computers and to avoid extremely large linearization weights described later,we add a thresholding operation in algorithm.Moreover,we use the classic Nesterov’s extrapolation technique to accelerate the convergence progress.A complete conver-gence analysis and a convergence rate of the AIS~3algorithm is established.Numerical experiments show that AIS~3takes the least time than other algorithms.