Iterative Algorithm Possible Problems

A very common problem in diverse areas of mathematics, physical sciences and system science consists of trying to find a point in the intersection of finite sets. This problem is referred to as the feasibility problem. This fundamental problem has many applications in and outside mathematics fields such as optimization theory, approximation theory, image reconstruction from projections and computerized tomography, control theory, signal and image processing, etc. Iterative projection algorithms have been recommended highly for solving this problem.By applying several accelerated techniques (relaxed technique, extrapolated technique, inertial technique) to projection methods (orthogonal projection, subgradient projection; approximate subgradient projection, perturbed projection), we present several algorithms for solving convex feasibility problem (CFP), split feasibility problem (SFP) and multiple-sets split feasibility problem (MSSFP). Furthermore several strong convergent algorithms by introducing some parameters sequences in infinite dimensions Hilbert space for solving the feasibility problems are constructed. Under some suitable conditions, their convergence are shown, numbers experiments manifest their feasibility. Thesis consists of several chapters.Chapter1recalls the research background of the feasibility problem.Chapter2recalls some preliminary concepts and relative results that will be used in the following chapters for solving feasibility problems. They are mainly convex analysis and monotone operators.Chapter3, Chapter4, Chapter5and Chapter6are the key contents of this thesis.In Chapter3. A parallel approximate subgradient projection algorithm and an accelerated parallel approximated subgradient projection algorithm for solving convex feasibility problem (CFP) are presented by substituting approximate subgradient projection for general projection. Compared with the existing approximate subgradient projection algorithm, the algorithms presented in this chapter use parallel process, i.e. at each iteration consider several approximation subgradient projections simultaneously, and adopt adaptive iterative technology which includes over-relaxed iterative process, so they can reduce the amounts of data storage and improve the convergence speed. And we also discuss the convergence of the methods under some conditions. Finally, the results of numerical experiments indicate that the parallel subgradient projection algorithms have faster convergence than that of the existing algorithm. Chapter4. We discuss an inertial orthogonal projection algorithm for solving the convex feasibility problem. Inertial Relaxed CQ algorithm and inertial perturbed projection algorithm are presented for solving the split feasibility problem. The inertial technique is simpler and more feasible than the general accelerated algorithm. Their asymptotic convergence is proved under some suitable conditions. Numerical examples are listed, which show that the proposed algorithms converge more quickly than the general algorithmsChapter5. Most of projection methods have only weak convergence in infinite Hilbert space for solving feasibility problems. For strong convergence, some extra assumptions on the sets such as compactness, finite dimensionality or uniform convexity are required. However, in most applications, these assumptions are not satisfied. In this chapter, we propose several strongly convergent projection algorithms for solving the convex feasibility problem and the split feasibility problem by introducing some parameter sequences, and prove their strongly convergent under some suitable conditions.Chapter6. The multiple-sets split feasibility problem (MSSFP), as a generalization of the split feasibility problem and a model for many inverse problems, is to find a point in the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be in the intersection of another family of closed convex sets in the image space. In this chapter, we present several accelerated algorithms by introducing extrapolated factors to solve the multiple-sets split feasibility problem. The convergence of the methods is investigated under some suitable conditions, numerical experiments are provided to illustrate the benefits of the extrapolation.Chapter7gives the conclusions and some open problems.