Algorithms And Convergence Analysis For The Split Variational Inequality Problem

Split variational inequality problem (SVIP) was introduced by Censor in2012. It is aproblem which combines variational inequality problem (VIP) with split feasibility problem(SFP). The SVIP has been used extensively in image signal reconstruction, sensor networks,radiation therapy treatment planning, resolution enhancement and so on. It has been concernedwidely by many domestic and foreign experts and scholars since being proposed. Some fruitfulresults have been achieved in its theories and algorithms. The main topic of the algorithmresearch on the SVIP in Hilbert space is its convergence analysis and working condition analysis.One hope to find an algorithm which converges much better and the working condition isordinarier. After Censor designed an algorithm for the SVIP, Moudafi modified and extended itso that the new algorithm can be used under weaker conditions. On this base, we give a furtherstudy on the algorithm designing, convergence analysis and using condition for the SVIP. Thefull thesis is divided into four chapters.The first chapter is an introduction. We describe the definition, application background andresearch situations of split variational inequality problem and analyze the significance andnecessity of studying it.In the second chapter, we study the split variational inequality problem in Euclidean space.Combining the Armijo step-size search method, we propose an extragradient method for solvingthe SVIP, which avoids the computation of the related spectral radius. The new algorithm weproposed expands the application scope of the methods for solving the SVIP.In the third chapter, we study the split variational inequality problem in Hilbert space. Wehope to design an algorithm in which the iterative sequence can converge strongly to a solutionof the SVIP under weaker condition. To achieve this goal, we design two algorithms: KM-likealgorithm and alternating projection algorithm. We can prove that the two algorithms mentionedabove have the good property that the iterative sequence generated by the algorithms convergestrongly to a solution of the SVIP under weaker condition. This expands the applied range of thesplit variational inequality problem.In the fourth chapter, we try to find the minimum-norm solution of the split variationalinequality problem in Hilbert space. Based on Tikhonov’s work for researching on solving the constrained linear system, the perturbation problem of variational inequality problem (VIP) isintroduced. Then, we use this work to design an algorithm for finding the minimum-normsolution of variational inequality problem. At last, we apply this algorithm to solve the splitvariational inequality problem and find a minimum-norm solution of it.