A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems

In this paper, we propose a hybrid nonlinear decomposition–projection method for solving a class of monotone variational inequality problems. The algorithm utilizes the problems’ structure conductive to decomposition and a projection step to get the next iterate. To make the method more practical, we allow solving of the subproblems approximately and adopt the constructive accuracy criterion developed recently by Solodov and Svaiter for classical proximal point algorithm and by the author for generalized proximal point algorithm. The Fejér monotonicity to the solution set of the problem is obtained by only assuming the underlying mapping is monotone and the solution set is nonempty. The parameter is allowed to vary in a larger interval than that of Auslender and Teboulle, and we also propose some improved self-adaptive strategies to choose the sequence of parameters, which makes the algorithm more flexible.