A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems. (English) Zbl 1039.47051

Yamada, Isao; Ogura, Nobuhiko; Shirakawa, Nobuyasu

Nashed, M. Zuhair (ed.) et al., Inverse problems, image analysis, and medical imaging. AMS special session on interaction of inverse problems and image analysis, New Orleans, LA, USA, January 10–13, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2979-3/pbk). Contemp. Math. 313, 269-305 (2002).

The aim of this paper is to show that (1) various algorithmic solutions to convexly constrained generalized inverse problems may be obtained in a unified manner from a certain hybrid steepest descent method, and (2) some particular versions of this method are gifted with notable robustness to the numerical errors. The obtained facts extend some recent contributions due to N. Ogura and I. Yamada [Numer. Funct. Anal. Optimization 23, 113–137 (2002; Zbl 1178.90346)].
For the entire collection see [Zbl 1003.00013].

Reviewer: Mihai Turinici (Iaşi)

MSC:

47J25	Iterative procedures involving nonlinear operators
47J07	Abstract inverse mapping and implicit function theorems involving nonlinear operators
47H10	Fixed-point theorems
90C25	Convex programming
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
90C30	Nonlinear programming

Keywords:

fixed point; nonexpansive map; convex projection; variational inequality; convexly constrained generalized inverse problem; hybrid steepest descent method; robustness; convergence analysis