Subgradient Extragradient Projection Algorithms For Solving Non-Monotone Variational Inequalities

The projection algorithm is an essential algorithm for studying variational inequality problems.In this thesis,we study a modified subgradient extragradient projection algorithm for variational inequalities in finite-dimensional Euclidean space.Firstly,the condition for the monotonicity of the mapping in the modified subgradient extragradient projection algorithm proposed by Malitsky and Semenov in 2014 is weakened to non-monotonicity.The global convergence of the algorithm is proved under the condition that the mapping is L-Lipschitz continuous and the solution set of the dual variational inequalities is non-empty.Furthermore,an adaptive subgradient extragradient projection algorithm is proposed by introducing an adaptive stepsize instead of a fixed stepsize.This algorithm does not need to know the Lipschitz constant of the mapping.Under the condition that the mapping is nonmonotone Lipschitz continuous and the solution set of dual variational inequalities is nonempty,we prove that the sequence generated by this algorithm converges to the solution of variational inequalities.Finally,by introducing the inertial technique,we propose an adaptive subgradient extragradient projection algorithm with inertial terms.Under the condition that the mapping is non-monotone Lipschitz continuous and the solution set of dual variational inequalities is nonempty,the global convergence of the algorithm is proved.Numerical experiments are given to show the effectiveness of the algorithm.