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Zbl 0883.47063
Censor, Y.; Reich, S.
Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. (English)
[J] Optimization 37, No.4, 323-339 (1996). ISSN 0233-1934; ISSN 1029-4945/e

Summary: A generalized ``measure of distance'' defined by $D_f(x,y):= f(x)- f(y)-\langle\nabla f(y),x- y\rangle$, is generated from any member $f$ of the class of Bregman functions. Although it is not, technically speaking, a distance function, it has been used in the past to define and study projection operators. In this paper, we give new definitions of paracontractions, convex combinations, and firmly nonexpansive operators, based on $D_f(x,y)$, and study sequential and simultaneous iterative algorithms employing them for the solution of the problem of finding a common asymptotic fixed point of a family of operators. Applications to the convex feasibility problem, to optimization and to monotone operator theory are also included.

MSC 2000:: *47H09 Mappings defined by "shrinking" properties
47H05 Monotone operators (with respect to duality)
90C25 Convex programming

Keywords: generalized distance; repetitive control; measure of distance; class of Bregman functions; projection operators; paracontractions; convex combinations; firmly nonexpansive operators; common asymptotic fixed point; convex feasibility problem; optimization; monotone operator theory

Cited in: Zbl 1156.47055

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