The Research On Projection Gradient Algorithms For Solving Variational Inequality Without Monotonicity

Variational inequality is an important part of optimization theory,which has been widely used in economy,management,finance,transportation and game theory.In recent decades,scholars have proposed a variety of iterative algorithms to solve variational inequality problems.However,these research mainly focus on solving monotone or pseudo-monotone variational inequality problems,and in the classical projection algorithm of variational inequality,the step size is related to the Lipschitz constant.Everyone knows that a large Lipschitz constant will lead to a small value of step size,further leading to a slow convergence rate.In this thesis,we improve the convergence conditions of the algorithm,under the non-monotonic assumption,we propose several projection gradient algorithms and obtain the convergence of the algorithms.By comparing with other relevant results through numerical experiments,it shows that the proposed algorithms are effective.The main arrangements are as follows:First of all,the research background and significance of variational inequality are briefly described,the research status of variational inequality at home and abroad is expounded,and some basic concepts and lemmas involved in this thesis are given.Next,we mainly discuss the extragradient projection algorithm for classical variational inequality problem.On the basis of the existing extragradient projection algorithm with inertia term,we improve the convergence conditions of the algorithm to be pseudo-monotone and uniformly continuous,and then prove that the algorithm converges weakly to the solution of variational inequality problem.Finally,numerical experiments show that the algorithm is applicative.After that,for non-monotone variational inequality problems,a modified inertial subgradient extragradient algorithm with Armijo linear search is given.Without the Lipschitz continuity assumption,the iterative sequence generated by the algorithm strongly converges to the solution of the variational inequality problem,numerical experiments show the effectiveness and advantages of the algorithm.At last,we extend the variational inequality to multi-valued variational inequality,and two kinds of modified projection algorithms are proposed.Under the assumption of non-monotone,we use adaptive step size to prove the convergence of the algorithm in finite dimensional space,and verify the effectiveness of the algorithms through numerical experiments.