Multi-step extragradient method with regularization for triple hierarchical variational inequalities with variational inclusion and split feasibility constraints

By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method with regularization, a hybrid multi-step extragradient algorithm with regularization for finding a solution of triple hierarchical variational inequality problem is introduced and analyzed. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a unique solution of a triple hierarchical variational inequality problem which is defined over the set of solutions of a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems (GMEP), finitely many variational inclusions, fixed point problems, and the split feasibility problem (SFP). We also prove the strong convergence of the proposed algorithm to a common solution of the SFP, finitely many GMEPs, finitely many variational inclusions, and the fixed point problem of a strict pseudocontraction. The results presented in this paper improve and extend the corresponding results announced by several others.