Inertial Gradient Methods For Variational Inequality Problems

Variational inequality problems are important part of nonlinear analysis,which has been widely used in engineering optimization,transportation and supply chain management.Projection algorithm is one of the effective algorithms for solving variational inequality problems.In this thesis,two kinds of modified inertial gradient projection algorithms are proposed based on the gradient projection algorithms in the real Hilbert space.The main work of this thesis are as follows:1.Two inertial subgradient extragradient algorithms with strong convergence have been proposed.Inspired by these algorithms,we improve the fixed constant stepsize into an adaptive stepsize,which enhances the convergence efficiency of these algorithms.Simultaneously these algorithms does not need to predict the Lipschitz constant.It promotes the application of these algorithms in practice.2.Combining the variational inequality problems and the fixed point problems to construct a new gradient projection algorithm to find its common solution.Moreover,based on the Censor’s algorithms,we generalize the non-expansive operator to the quasi-nonexpansive operator.Then find the common solution of fixed point problems and variational inequality problems.3.The monotonicity assumption of the operator have been weaken,constructs a new algorithm under the condition that the operator is pseudo-monotone and Lipschitz continuous.It have been proved that these algorithms weakly converges to the common solution of the variational inequality problems and the fixed point problems.