Research On Proximal-type Algorithms For Split Variational Inclusion Problems

Split variational inclusion problem is one of the important content in the field of nonlinear programming,and has provided a framework for many mathematical problems,such as split feasibility problem,proximal split feasibility problem,linear inverse problem,split variational inequality problem and so on.In this framework,proximal-type algorithms effectively solve some practical problems,such as signal recovery,image denoising,sensor networks and so on.Hence,the research on proximal-type algorithms for solving split variational inclusion problems is of an important theoretical significance and application value.Since the stepsizes of the classical proximal-type algorithms depend on the bounded linear operator norm,which is difficult to compute,so strategies for selecting the stepsizes,such as the Armijo-type linesearch rule and self-updated stepsize methods,have been proposed.The proposed strategies require the algorithm to compute the proximity operator multiple times,which affects the effectiveness of the algorithms.In real Hilbert space,their convergence proof does not involve some conditions,such as Lipschitz continuity of the monotone operators and firm-nonexpansiveness of the proximal mappings.From these considerations,in this paper,these conditions are weakened,we introduce several simply and effectively split proximal algorithms.The specific work is the following:1.Inertial split proximal algorithms for solving split variational inclusion problems with non-Lipschitz mappings are studied.(1)For new stepsize algorithms proposed by Tang et al,the setting range of original parameters is widened.A non-Lipschitz stepsize strategy is introduced,and the disadvantage that the original stepsize tends zero is overcome.Two optimization approaches are proposed by using the inertial method.Compared with the original algorithms,the proposed algorithms only compute the proximity operator twice for each iteration,thus improving the computational efficiency of the algorithms.For the inertial projection contraction algorithm proposed by Tong et al.,the stepsize rule for the the Armijo-type linesearch method is modified,and the strong convergence of the new algorithms is proved under relatively weak conditions.(2)As their applications,new algorithms effectively solves image deblurring,split feasibility and split minimization problems.(3)In order to obtain strong convergence of the above algorithm,an inertial Halpern-type CQ algorithm is proposed by using the Halpern method,and the effectiveness of the new algorithm is verified by an example.2.We study inertial relaxed CQ algorithms for split feasibility problems with non-Lipschitz gradient operators.(1)The inertial relaxed CQ algorithm considered by Sahu et al is modified.First,the inertia used by kesornprom et al is skillfully used for replacing the original inertia,which not only simplify the convergence proof of the modified algorithm,but also shows a new proof way.Second,a new stepsize is non-monotone and bounded.the proposed stepsize avoids the drawback that the original stepsize fast tends to zero.This helps improve the efficiency of the original algorithm.(2)Furthermore,the definition of stepsize is reasonable without Lipschitz continuity of the gradient mapping.After removing firm-nonexpansiveness of the projection mappings,we obtain weak convergence of the modified algorithm.Also,we extend our modified method to the multiple-sets split feasibility problems.Finally,numerical examples demonstrate the effectiveness of the modified algorithm.(3)Based on the modified algorithm,we use the inertial effect and Halpern iteration scheme to propose an inertial Halpern-type CQ algorithm.Meanwhile,we establish the strong convergence of the proposed algorithm.Numerical results disclose the effectiveness of the proposed algorithm.3.We study an inertial split proximal algorithm for solving proximal split feasibility problems with non-expansive mappings.(1)By weakening firm-nonexpansiveness of the proximal mappings,a new stepsize is introduced,the inertial split proximity algorithm of Shehu et al is modified by using the inertial method and viscosity algorithm.(2)Under Lipschitz continuity of the gradient mapping assumption,strong convergence of the new algorithm is established.Numerical experiments verify the effectiveness of the new algorithm.