A Projection Algorithm For Pseudomonotone Vector Fields With Convex Constraints On Hadamard Manifolds

In this thesis,a new Riemann projection method is proposed for finding a zero of a pseudomonotone vector field with a convex constraint on a Hadamard manifold.The search direction of this new method is constructed as a speical linear combination of the value of the vector field and the direction vector between the adjacent iterative points.By using a line search technique and the hyperplane projection method,a predicted iterative point can be obtained.The new iterative point is defined as the projection of this point onto the constraint set.The global convergence property of this algorithm is established under the assumptions that the constructed halfspace is closed and convex,the tangent vector field is continuous,and the solution set is nonempty.Numerical experiments show the e ciency of this new derivative-free iterative method.The first chapter is the introduction,which summarizes the research status of the vector field zero-point solution and the main work of this thesis.The second chapter is the preliminary knowledge.This chapter introduces some basic concepts and important conclusions on Hadamard manifolds.In addition,we review the extragradient method for finding a zero of a pseudomonotone vector field on Hadamard manifolds,and Korpelevich’s method for solving variational inequality corresponding to pseudomonotone vector fields on Hadamard manifolds.The third chapter is divided into five sections.the first section is the description of a zero point problem of a pseudomonotone vector field with a convex constraint on Hadamard manifolds.The second section presents a Riemannian modified projection method for solving the problem.The third section analyzes and proves the global convergence of the algorithm.Section IV presents some numerical experiments.Section five gives a summary of this chapter.In Chapter 4,this chapter is divided into three sections.The first section gives the application of the proposed new Riemannian projection algorithm in Euclidean space,and the second section gives the local convergence speed analysis of the corresponding Euclidean space projection method.The last section gives a summary of this chapter.Chapter 5 summarizes the full text,explaining further work to be carried out in the future and possible future research directions.