On Numerical Algorithms For Several Convex Feasibility Problems

Many problems in management science,automation control and mechanics can be transformed into the convex feasibility problems,which consist of finding an element in the intersection of two or more closed and convex sets.As the continuous development of interdisciplinary,convex feasibility problems play an increasingly important role in diverse fields,such as computer science,transportation,engineering technology,signal processing and so on.Variational inequality problems,monotone inclusion problems and fixed point problems are important branches of convex feasibility problems.Those three problems have very closely relationships,since they can transform mutually.Besides,variational inequality problems,monotone inclusion problems and fixed point problems have extensive real applications.The thesis aims to investigate several effective approximation algorithms in various spaces with their applications in concrete examples.Three aspects including their algorithm designs,the convergence analysis and numerical implementations,are investigated.The obtained results extend and improve some known results.The thesis is divided into eight chapters,which are described as follows:Chapter 1.We introduce the domestic and foreign research status of the convex feasibility problem,the main work and the structure arrangement of this dissertation and recall some preliminaries,which will be required in the following chapters for solving convex feasibility problems.Chapter 2.We construct a modified inertial subgradient extragradient algorithm for finding the solution of the variational inequality problem.Under the mild assumption that the underlying mapping is sequentially weakly continuous,pseudomonotone and Lipschitz continuous,we prove the weak convergence of the sequence generated by the proposed method.Numerical results show that our new algorithm has a faster rate of convergence and a better approximation accuracy,in comparison with some existing algorithms.Chapter 3.Two modified iterative algorithms are suggested for solving pseudomonotone variational inequality problems on the basis of the inertial Tseng algorithm.Strong convergence theorems are established under some suitable assumptions.The proposed algorithms have the advantage of the small computational effort since they only calculate the projection operator once per iteration.By adopting the Armijo stepsize research rule,the proposed algorithms converge strongly even without the restriction on the Lipschitz constant of the underlying mappings.Our algorithms are applied to solve fuzzy convex programming problems and numerical examples are given to support theoretical results.Chapter 4.A three-steps hybrid iterative algorithm is given to compute the approximate solution of a double hierarchical inequality problem.The double hierarchical inequality problem,namely,the variational inequality problem is defined over the solution set of another variational inequality problem,is investigated.Furthermore,by utilizing the proposed algorithm,the corresponding neural network model is given.The obtained algorithm is applicable for solving utility maximization based network bandwidth allocation problem.Numerical results show that the proposed algorithm converges more quickly than that of some existing algorithms.Chapter 5.A multiple-steps hybrid iterative algorithm is constructed for solving a multiple-sets maximal monotone inclusion problem via the combination of the forwardbackward splitting method,Tseng method and the inertial technique.The strong convergence theorem is obtained under certain assumptions.The numerical experiment is given to indicate that the new algorithm is valid and applicable to the signal recovery problem.Chapter 6.In the setting of Banach spaces,by combining the viscosity method and the Bregman projection method,a Harlpern-type projection iterative algorithm is constructed for approximating the fixed point of a semigroup of Bregman quasi-nonexpansive mappings.The strong convergence of this algorithm is proved under the assumption that the solution set is nonempty.Numerical experiments are done to verify the effectiveness and feasibility of the theoretical results.Chapter 7.A modified variable metric forward-backward splitting algorithm is proposed with error items,for finding a common element of the solution set of a monotone inclusion problem and the solution set of zeros of an inverse strong monotone mapping.A hybrid implicit and explicit iterative scheme is proposed with error items,for finding an element in the intersection of the fixed point solution set of a finite number of nonexpansive mappings and the solution set of a zero problem.The weak and strong convergence of the proposed algorithms are proved respectively,under some different conditions.Chapter 8.We give the conclusion of the research results of this thesis,and prospect the future research.