An Improved Projection Algorithm For Pseudo-monotone Variational Inequality Problems In Banach Spaces

Variational inequality is an important nonlinear problem,which is widely used in the fields of network economics,engineering management,macro-control and so on.It has become an important tool in the study of physics,mathematics and economics.There are many methods to solve monotone variational inequality problem,among which the projection method is a convenient and effective method.Because the projection computation does not require the function to be derivable,the projection method has been paid more and more attention and become a very important method.In this paper,a class of inertial projection algorithms and adaptive projection algorithms for solving pseudo-monotone variational inequalities are proposed in Banach space.Their convergence is proved under appropriate assumptions and the results of numerical experiments are analyzed.The main contents of the full text are as follows:Firstly,by combining the inertia term and the new Armijo step size rule,an inertial subgradient external gradient algorithm is proposed to solve variational inequalities in Banach space.The strong convergence of the algorithm is proved when the mapping satisfies pseudo-monotone,uniformly continuous and weakly continuous sequence.At the same time,numerical experimental results of the algorithm are given,which show the effectiveness of the algorithm.Secondly,an inertial projection algorithm is proposed to solve variational inequalities in Banach space by combining the inertial term and constructing a new adaptive step size.The weak convergence of the algorithm is proved under the conditions that the mapping satisfies pseudo-monotone,Lipschitz continuous,sequence weakly continuous and Lipschitz constant is not required.At the same time,numerical experimental results of the algorithm are given,which show the effectiveness of the algorithm.