Extragradient Methods And Proximal Point Algorithms For Nonmonotone Variational Inequality Problems

The extragradient method and the proximal point algorithm are two classical and effective algorithms for solving variational inequalities.So far,the convergence of the extragradient method and the proximal point algorithm requires the underlying mapping to be at least pseudomonotone and continuous.In order to further broaden the application scope of these two algorithms,we weaken the pseudomonotonicity of the underlying mapping,and design the extragradient method and the proximal point algorithm for nonmonotone variational inequalities.Our main contributions are as follows:(1)An extragradient method for nonmonotone variational inequalities is introduced in this thesis(without assuming any generalized monotonicity).In order to ensure the global convergence of the new extragradient method,in addition to the basic assumption of the continuity of the underlying mapping,we only need to assume that the dual variational inequality has a solution.On the one hand,if the variational inequality has a solution and the mapping is pseudomonotone,then the dual variational inequality has a solution.Therefore,our algorithm applies to pseudomonotone variational inequalities.On the other hand,if the variational inequality has a nontrivial solution and the mapping is quasimonotone,then the dual variational inequality has a solution.This shows that our algorithm can be applied to a large class of quasimonotone variational inequalities.This is an important contribution to the traditional extragradient method.However,the examples in this thesis show that even if the dual variational inequality has a solution,the mapping is not necessarily quasimonotone,let alone pseudomonotone.This implies that the new extragradient method can be applied to nonmonotone variational inequality problems.Two numerical examples are given to demonstrate the effectiveness of the algorithm.(2)A proximal point algorithm for nonmonotone variational inequalities is introduced in this thesis.We first prove an equivalent form of the exact proximal point algorithm.Then,based on this equivalent form,the exact and inexact proximal point algorithms for nonmonotone variational inequalities are designed in this thesis.In order to ensure the global convergence of these algorithms,we only need to assume that the dual variational inequality has solutions and the mapping is continuous.It is worth noting that this is a very weak condition to guarantee the global convergence of the proximal point algorithm,and no generalized monotonicity is assumed.In addition,the examples in this thesis show that our algorithms can be applied to nonmonotone variational inequalities.Finally,an application example of the algorithm for quasiconvex optimization is given,and the effectiveness of the algorithm is verified by a numerical example.