Research On Approximation Algorithm Of The Variational Inequalities

As the main promotion of the variational principle, variational inequalities have close contact with other disciplines and own wide application prospect. Recently, in order to overcome the problem that it is difficult and unnecessary to make the precise iterate calculation, the approximation algorithm of variational inequalities have received wide attention and become a great focus of research.Based on the analysis of inexact proximal point algorithm, the paper constructs a subgradient half space for general variational inequality which contains the original projection domain and depends on the iterative format of extragradient algorithm in the condition of pseudo monotone, then the approximate proximal extragradient algorithm is put forward. Further, for the classical variational inequality, it is proved the algorithm is both applicable for the monotone operator and pseudomonotone operator, which expands the operator constraints; Then the paper combines the half space approximate projection with alternating direction method, and constructs iterative under the half space projection, further proposed the improved approximate alternating direction method and proves the global convergence of the algorithm. Finally, combined with the above two algorithms, and based on the equivalent deformation of double-objective programming and variational inequalities, due to the half space projection, it is proposed the approximate proximal extragradient algorithm for double-objective programming.