Research On Algorithms And Applications For Some Split Feasibility Problems

As an important inverse problem model,the split feasibility problem is widely used in real-world problems,such as image processing,intensity-modulated radiation therapy,signal recovery,electricity production and so on.Based on the background of fixed point theory and optimization theory,the split feasibility problem has been widely studied and has arised many important split problems.In this dissertation,the split feasibility problem,the split common fixed point problem and the split monotone variational inclusion problem will be investigated in the framework of infinite-dimensional Hilbert spaces.Based on the existing algorithms and theories,some efficient and practical algorithms are proposed to approximate the solution of the research problems on the aspects of algorithm structure,convergence analysis and parameter constraints.The main contents of this dissertation are as follows:Firstly,in infinite-dimensional Hilbert spaces,the split feasibility problem is stud-ied by using the extragradient method and the projection and contraction method in the variational inequality problem.Based on the Armijo linesearch stepsize criterion and the monotone adaptive stepsize criterion,a relaxed alternated inertial extragradient algorithm and a relaxed alternated inertial projection and contraction algorithm are proposed.Under some appropriate conditions,weak convergence theorems of the proposed algorithms are established.The level sets of convex functions are used as closed convex sets,so that the projection mappings in the relaxed algorithms have closed forms.Moreover,under the action of the alternated inertial method,the proposed algorithms retain the acceleration effect in the inertial form and weaken the restriction on the inertial extrapolation term co-efficient.Finally,numerical results in the signal recovery problem demonstrate that the proposed algorithms are effective and outperform the comparative algorithms.Secondly,in infinite-dimensional Hilbert spaces,the split common fixed point prob-lem for demicontractive mappings is studied.Based on the inertial technique,this dis-sertation proposes an inertial viscosity algorithm with a Meir-Keeler contraction mapping and an inertial hybrid projection algorithm.By constructing half-spaces with closed form and its associated projection mappings,the inertial hybrid projection algorithm reduces the constraints on the inertial parameters.The proposed algorithms include a new adaptive stepsize and effectively avoid estimation of the bounded linear operator norm.Without involving semicompactness of the demicontractive mapping,the iterative sequences gen-erated by the two algorithms converge strongly to the solution of the split common fixed point problem.The strong convergence properties are also applied to the split common fixed point problem in the case of the strictly pseudocontractive mapping,the directed mapping and the firmly nonexpansive mapping.Numerical results show that the two al-gorithms are effective and improve the existing algorithms.Thirdly,in infinite-dimensional Hilbert spaces,the split monotone variational inclu-sion problem is studied and two inertial hybrid steepest descent algorithms are proposed.The proposed algorithms effectively avoid the calculation of the bounded linear opera-tor norm by means of an adaptive stepsize criterion.Under some appropriate conditions,the hybrid steepest descent method involving the Lipschitz continuous mapping and the strongly monotone mapping ensures that the iterative sequences constructed by the pro-posed algorithms converge strongly to the solution of the split monotone variational inclu-sion problem,which is also the unique solution of a specific variational inequality.The strong convergence theorems are also applied to the split variational inclusion problem,the split variational inequality problem,and the split minimization problem.Meanwhile,the hybrid steepest descent algorithms also include other forms of strongly convergent algorithms,such as the viscosity algorithm,the Halpern algorithm and the Mann-type al-gorithm.Finally,numerical examples in finite-dimensional spaces and L₂spaces verify the effectiveness of the proposed algorithms.