Some New Subgradient Extragradient Algorithms For Solving Monotone Variational Inequalities

In this thesis,we mainly study some new subgradient extragradient algorithms of monotone variational inequalities in Hilbert space.By constructing some half-spaces and linesearch conditions,we obtain the strong convergence of the algorithms in Hilbert space.The thesis is divided into three chapters,and the specific content is as follows:In chapter 1,we introduce the research background,current research status at home and abroad,and the basic definitions and lemmas necessary for the research.In chapter 2,we propose two improved subgradient extragradient algorithms.Different from the general subgradient extragradient algorithms which need projection onto a general non-empty closed and convex subset,in our new algorithms,we only need projection onto some specific half-spaces,which is easier calculation.At the same time,by a linesearch condition,the strong convergence of the algorithms are proved with the Lipschitz constant of the mapping1)is unknown in Hilbert space.Finally,numerical experiments show that algorithms proposed in this chapter have better convergence effect than other known algorithms in some cases.In chapter 3,we propose a like-subgradient extragradient algorithm.In our algorithm,we only need projection onto two specific half-spaces rather than some general non-empty closed and convex subsets.Under the assumption that the mapping 1)is monotone and Lipschitz continuous(Lipschitz constant is unknown),we can get the strong convergence result of the algorithm in Hilbert space with using a line search condition and combining a strict contraction mapping at the end of the algorithm.Finally,the corresponding numerical experiments are given at the end of this chapter.