On Algorithms For The Split Feasibility Problem And Its Extension Problems

The split feasibility problem (SFP) is one of the important research problems in the field ofoptimization. It has been not only used in signal processing and image reconstruction widely, butalso in system identification, economy, military field and so on. It has been concerned widely bymany domestic and foreign experts and scholars since being proposed. Some fruitful results havebeen achieved in its theories and algorithms. In the early time, the influential algorithm to solveit is CQ-algorithm which avoids calculating the inverse of matrix. Meantime it also brings newproblems inevitably. So many scholars did further research on CQ-algorithm and achievedfruitful results. Relaxing closed convex sets C and Q through constructing the half space to solvethe split feasibility problem become more popular later. But the existing algorithms have tocalculate spectral radius of matrix, estimate Lipschitz coefficient or do line research in theprocess of getting the suitable step size which is often difficult to implement in practice. In thispaper, on the basis of the predecessors, we improve this insufficient such that the step size can becalculated directly.Due to the need of practice, Censor put forward an extension problem of the split feasibilityproblem: multiple-sets split feasibility problem.This extension problem attracted many scholarsto study and they achieved fruitful results. After further research on split feasibility problem,Byrne and Moudfi put forward its general form, namely, the split equation problem. Manyscholars observe it from different aspects and so another extension problem of the split feasibilityproblem arises, that is, the minimum2-norm solution of the split feasibility problem. The fulltext is divided into four chapters.The first chapter is an introduction. We state the origin of the split feasibility problem, theapplication background of it and its research status.In the second chapter, on the basis of predecessors, we first put forward CQ-like algorithmand give its convergence analysis. The algorithm avoids the calculation of spectral radius ofmatrix, estimating Lipschitz constant and saves calculation time. Secondly, we give a memorygradient projection algorithm and its convergence proof. The new algorithm uses two points anda parameter to get the next point instead of one point which increases the flexibility of thealgorithm. At the same time, through relaxing the closed convex sets C and Q, we propose therelaxed form of the memory gradient projection algorithm which is more easier in calculatingthe related projections.In the third chapter, we mainly discuss two expanded forms of the split feasibility problem: the multiple-sets split feasibility problem and split equation problem. A sequence projectionalgorithm for solving the multiple-sets split feasibility problem and a simple projection algorithmfor solving the split equation problem and its relaxed form are put forward. Meanwhile we provethe convergence of these algorithms.In the fourth chapter, on the basis of optimization theories, we give a gradient algorithm toget the minimum2-norm solution of the split feasibility problem. The global convergence of thealgorithm is also given.