Several Methods For Solving Pseudomonotone Variational Inequalities

The theory of variational inequalities is one of the effective methods for studying various problems in the field of nonlinear analysis.The solution of variational inequality essentially characterizes a set of points in a space that satisfies this inequality.Its existence and uniqueness have been discussed in depth and detail.Because many problems in the field of nonlinear analysis can be described by variational inequalities,solving these problems can be classified as solving variational inequality problems.Currently,through the research of many outstanding scholars on numerical solutions of variational inequalities,classic and effective algorithms such as contraction algorithm,relaxation algorithm,projection algorithm,etc.have emerged.This thesis mainly uses classical projection algorithms,viscosity algorithms,inertia algorithms,etc.to study the approximation problem of solutions to pseudo monotonic variational inequalities.Firstly,most research on variational inequality problems in Hilbert spaces has focused on strongly monotone or monotone operators,resulting in weak convergence of sequences.And the step size of the algorithm depends on the Lipschitz constant or line search,which is not easy to obtain.The line search determines the step size through an internal loop.So this thesis proposes an adaptive viscosity inertial algorithm to solve pseudo monotone variational inequality problems.By combining the ideas of inertial and viscosity,an appropriate iterative algorithm is constructed to accelerate the approximation of solutions to pseudo monotone variational inequalities.This thesis not only proves the strong convergence of the constructed algorithm,but also compares it with numerical examples to demonstrate the feasibility and superiority of the proposed theoretical results.Secondly,there are not many achievements in solving pseudo monotone variational inequality problems in Banach spaces,and the convergence rate of the algorithm still needs to be improved.This thesis proposes using Bregman distance projection in first-order methods,combined with viscosity and inertial ideas,to construct a new algorithm using adaptive step size to accelerate the approximation of solutions to pseudo monotone variational inequality problems in Banach spaces.In addition to conducting strong convergence analysis on the algorithm,numerical examples were also used for comparative analysis to emphasize the effectiveness and feasibility of the algorithm.Finally,this thesis extends the solution of fixed point problems for contraction operators in Banach spaces to the common solution problem for solving variational inequalities for pseudo contraction operators and accretive operators,constructs a viscosity iterative approximation algorithm,and establishes a series of strong convergence theorems.