The Improved Subgradient Extragradient Algorithms And Partially Combination Projection Method

Under the framework of Hilbert space,we propose and study totally relaxed,self-adaptive subgradient extragradient algorithms and partially combination projection algorithm for solving variational inequalities problems.In view of Lipschitz continuous and monotone variational inequalities,Censor proposed the subgradient extragradient algorithm.In each iteration,this algorithm replaces the projection defined on the field of definition in correction iteration step by the projection on a constructed half-space,this algorithm improve the extragradient algorithm and is easy to implement.Next,we put forward totally relaxed,self-adaptive subgradient extragradient algorithm,In every iteration,two projections on the definition domain both in prediction and correction steps are replaced respectively by two projections on some constructed half-space with special structure(when the domain is intersection of finite level sets,it is replaced by the projection onto the intersection of finite half-spaces).Thus,the feasibility of subgradient extragradient algorithm is further improved.In addition,between our algorithms,iteration parameters can be selected in a adaptive way,without computing or estimating the Lipschitz constant of operators.This is another advantage of our new algorithms.According to the two cases of domain for single convex function level set and the intersection of finite convex functions level sets,we design totally relaxed,self-adaptive subgradient extragradient algorithms and prove the weak convergence theorems.Also,we carry out a data simulation test for our algorithms and the numerical results showed the superiority of them.For Lipschitz continuous and strongly monotone variational inequalities,which are defined on the intersection of finite convex functions level sets.Most algorithms usually need to implement successive projection in every iteration step,which makes the algorithm so complex that it may have a larger computation workload.In order to overcome these weaknesses,we propose the partially combination projection algorithm.The core is to choose a plurality of convex functions to construct a half-space according to certain rules,the iterative scheme only involves the calculation of projections on the half-space.In this way,the calculation format is simple and the devised algorithm is easy to achieve.Strong convergence of above algorithm is proved,its numerical results also show its advantages.