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The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. (English) Zbl 1013.49005
Butnariu, Dan (ed.) et al., Inherently parallel algorithms in feasibility and optimization and their applications. Research workshop, Haifa, Israel, March 13-16, 2000. Amsterdam: North-Holland/ Elsevier. Stud. Comput. Math. 8, 473-504 (2001).

This paper presents a simple algorithmic solution to the variational inequality problem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space. The algorithmic solution is named the hybrid steepest descent method and generates a sequence strongly convergent to the solution of the problem.

The applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problem.

MSC:
 49J40 Variational methods including variational inequalities 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J20 Inequalities involving nonlinear operators 65K10 Optimization techniques (numerical methods) 47H09 Mappings defined by “shrinking” properties