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Abstract
In this paper, we consider, in a finite dimensional real Hilbert space , the variational inequality problem VIP
: find
, where
is nonexpansive mapping with bounded
and
is paramonotone and Lipschitzian over
. The nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping is a typical example of such a variational inequality problem. We show that the hybrid steepest descent method, of which convergence properties were examined in some cases for example (Yamada, I. (Citation2000). Convex projection algorithm from POCS to Hybrid steepest descent method. The
Journal
of
the
IEICE (in Japanese) 83:616–623; Yamada, I. (Citation2001). 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. Elsevier; Ogura, N., Yamada, I. (Citation2002). Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23:113–137), is still applicable to the case where
and T satisfy the above conditions.