A hybrid steepest-descent method for variational inequalities in Hilbert spaces

Abstract

A Mann-type hybrid steepest-descent method for solving the variational inequality ⟨F(u*), v − u*⟩ ≥ 0, v ∈ C is proposed, where F is a Lipschitzian and strong monotone operator in a real Hilbert space H and C is the intersection of the fixed point sets of finitely many non-expansive mappings in H. This method combines the well-known Mann's fixed point method with the hybrid steepest-descent method. Strong convergence theorems for this method are established, which extend and improve certain corresponding results in recent literature, for instance, Yamada (The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Amsterdam, Holland, 2001, pp. 473–504), Xu and Kim (Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theor. Appl. 119 (2003), pp. 185–201), and Zeng, Wong and Yao (Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theor. Appl. 132 (2007), pp. 51–69).