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).

Keywords:

• Mann-type hybrid steepest-descent method
• variational inequality
• non-expansive mappings
• Hilbert space
• demiclosedness principle
• strong convergence

AMS Subject Classifications:

• 47H09
• 47J20

Acknowledgements

Lu-Chuan Ceng was supported in part by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation of Shanghai. Jen-Chih Yao was Supported in part by a grant from the National Science Council.