Projection Algorithms For Solving Nonmonotone Variational Inequalities

In this thesis,projection algorithms for solving nonmonotone variational in-equalities is studied in the finite dimensional Euclidean spaces R⁹).Firstly,we propose a half-space projection algorithm for solving nonmonotone variational inequalities,which constructs a projection half-space that is different from the known methods.Under the assumption that the solution set of its dual variational inequality is nonempty,we prove the global convergence of the new algorithm.Numerical experimental results show that the new algorithm is effective and feasible.Secondly,we propose a self-adaptive single-projection algorithm for solving nonmono-tone variational inequalities.This algorithm uses a self-adaptive technique to calculate the step-size,overcoming the disadvantage of using Armijo line search procedure to search for step-size in each iteration in known literature,which requires projection to the feasible set.Each iteration of the new algorithm only requires one projection to the feasible set.Under the assumption that the solution set of its dual variational inequality is nonempty,we prove the global convergence of the new algorithm.To verifies the effectiveness and feasibility of the new algorithm,some numerical results are provided.Finally,we propose a modified projection and contraction algorithm and a modified Tseng-type extragradient algorithm for solving nonmonotone variational inequalities.By improving generation method of the next iteration point in the projection and contraction algorithm and the Tseng-type extragradient algorithm,we do not need to assume any generalized monotonic-ity when proving the global convergence of these two new algorithms.Under the assumption that the solution set of its dual variational inequality is nonempty,we prove the global con-vergence of these two new algorithms.Numerical experiments of the algorithm are given,and the numerical experimental results show that the two new algorithms can be applied to solving nonmonotone variational inequality problems.