Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings

The hybrid steepest descent method is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to broad range of convexly constrained nonlinear inverse problems in real Hilbert space. In this paper, we show that the strong convergence theorem [Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications. Elsevier, pp. 473–504] of the method for nonexpansive mapping can be extended to a strong convergence theorem of the method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings called quasi-shrinking mapping. We also present a convergence theorem of the method for paramonotone variational inequality problem over the bounded fixed point set of quasi-shrinking mapping. By these generalizations, we can approximate successively to the solution of the convex optimization problem over the fixed point set of wide range of subgradient projection operators in real Hilbert space.