Research On Projection Algorithms And Their Applications For Solving Variational Inequalities

In this paper,three projection algorithms for variational inequalities are proposed under two different preconditions on the mapping F.Descent properties of the search direction and the global convergence of the three algorithms are proved in this article.In the first chapter,some basic knowledge and properties of variational inequalities and projection mapping are given.The known projection algorithms and numerical experiments for solving variational inequalities need to be used in this paper are briefly introduced,and the corresponding analysis will be carried out in the final numerical experiments.In the second chapter,under the condition that F is monotone,a new search direction for solving variational inequality problems is proposed.And the new search direction function is a convex combination of the known directions.When the iterative points sequence generated by the new algorithm converge to the solution of the variational inequality,the search direction function does not converge to zero,and the distance from the iteration point to the solution of variational inequality is strictly monotonically decreasing.The global convergence of the algorithm is proved under the conditions that mapping F is monotone and continuous.In the third chapter,under the condition that F is strong monotone,two new search directions for solving variational inequality problems are proposed.First,under certain conditions,a search direction is given.The base is that the solution of the variational inequality come from the complementary problem is equivalent.Secondly,on the basis of the direction,another new projection direction is given based on the appropriate combination of the current search direction and the previous search direction.The distance from the iteration point to the solution of variational inequality is strictly monotonically decreasing.The global convergence of the algorithm is proved under the conditions that mapping function F is strong monotone and Lipschitz continuous.In the fourth chapter,three numerical experiments about the variational inequalities are given,which show the three projection algorithms are feasible in solving the variational inequalities.And efficiency has also been improved for solving the variational inequalities.