On Extragradient Algorithms For Variational Inequalities And Related Problems

The variational inequality problem has attracted the attention of mathematicians,economists and engineers since it was put forward,and it has become an effective tool to study theoretical models such as mathematical models,economic management models and engineering physics models.At the same time,the variational inequality problem is widely used.It covers many fields such as operations research and optimization,supply chain management,information engineering,and transportation.Scholars mainly conduct research on variational inequality problems in terms of algorithms and applications.This thesis summarizes the projection iterative methods for variational inequality problems at domestic and international,and uses the fixed point theory as a tool to construct a new numerical iterative method for solving numerical solutions of variational inequality problems.Common solutions for solving fixed point problems and variational inequalities in Bert spaces and Hadamard manifolds.In Chapter 2,we construct some new extragradient projection algorithms in Hilbert space.Based on the algorithms of Tseng and Thong,this thesis constructs three iterative algorithms using the fixed point theory and method.The first iterative algorithm is based on the Thong's method,which modifies the constant step size to an adaptive step size;The second iterative algorithm introduce subgradient projection and contractive operators in the iterative format of Tseng's method;the third iterative algorithm introduces subgradient projection and Mann-like iterative format based on the Thong's acceleration method.In Chapter 3,we propose a new subgradient extragradient projection algorithm in Hilbert space.This thesis summarizes the research work of Iusem,Migorski and Ishikawa,and uses the fixed point theory and method as a tool to construct a strong convergent iterative algorithm.This algorithm can solve both pseudo-contractive fixed point problems and pseudo-monotone variational inequality problem.In Chapter 4,we give the new subgradient extragradient projection algorithm in the Hadamard manifold.Based on the research results of Ishikawa and Konrawut,this thesis constructs a strongly convergent iterative algorithm using the fixed point theory and method.The algorithm combines the Ishikawa's method for solving fixed point problems and the Konrawut's method for solving variational inequality problem,so that it can be solve the common solutions of quasi-nonexpansive fixed point problems and pseudo-monotone variational inequality problem.