An Inexact Extragradient Method For Variational Ineqaulity

The variational inequality problems (VIPs for short) are an important topic in Operations Research, and the Korpelevich extragradient method is one of the most fundamental methods for VIPs. So far, the Korpelevich extragradient method has been always studied in exact cases. However, computational errors are always unavoidable in each function evaluation. In this paper, we give a new iterative formFurthermore, the convergence of this inexact method with absolute error (?)is proven under the assumption that X is a bounded closed convexset and F is a pseudomonotone and Lipschitz continuous mapping. And the convergence of this inexact method with relative error (?), where(?) andλ_k,μ_k≥0, proven under the assumption that F is apseudomonotone and Lipschitz continuous mapping.The general variational inequality problems (GVIPs for short) are an important generalization of the classical VIPs, so it has more applications. In the third chapter, we generalize the inexact Korpelevich extragradient method for classical VIPs to GVIPs, and also show the convergence of this inexact method.The fourth chapter is relatively independent of other chapters, it mainly study aclassical method——the proximal point method for maximal monotone inclusions. Wegive a new proof for the convergence rate of the inexact proximal point algorithm, and our technique seems simpler than the original one.