Inertial Projection Algorithms For Variational Inequalities And Fixed Point Problems

In this thesis,we consider two projection algorithms with inertia terms for variational inequalities and fixed point problems in real Hilbert space.We also prove the convergence of these methods under some mild conditions.Firstly,we propose an inertial subgradient extgradient algorithm for variational inequality in Hilbert space,whose feasible set is defined by level set of smooth function.Combining inertial acceleration techniques,the algorithm just need do two half space projections,and use only one value of objection function per iteration.Suppose that the objective function is monotonicity and Lipschitz continuous,The weak convergence of the algorithm is proved.Then,we propose an inertial subgradient extragradient algorithm with line search for solving the common solution of variational inequality and fixed point problems,where we use the Mann-type iteration format and inertial acceleration technique and line search.The algorithm just need do two half space projections per iterationt.A notable advantage of new algorithm is that we don’t need know the exact value of Lipschtiz continuity coefficient of objective function.Under the assumption that the mapping is monotonicity and Lipschitz continuous,and Fixed point mapping is a quasi-nonexpansive mapping,the weak convergence of the algorithm is proved.Last,We also present some numerical experimental results of two algorithms and compare them with some existing projection methods to show its efficiency.