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Zbl 0913.47048
Deutsch, Frank; Yamada, Isao
Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.
(English)
[J] Numer. Funct. Anal. Optimization 19, No.1-2, 33-56 (1998). ISSN 0163-0563; ISSN 1532-2467/e

Let $T_i$ $(i= 1,2,\dots, N)$ be nonexpansive mappings on a Hilbert space ${\cal H}$, and let $\Theta:{\cal H}\to \bbfR\cup \{\infty\}$ be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of $T_i({\cal H})$ for some $i$. The authors prove that $\Theta$ has a unique minimum over the intersection of the fixed point sets of all the $T_i$'s at some point $u^*$. Then a cyclic hybrid steepest descent algorithm is proposed and it is shown that it converges to $u^*$.
[J.Appell (Würzburg)]
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
90C25 Convex programming
65K05 Mathematical programming (numerical methods)
65K10 Optimization techniques (numerical methods)
90C30 Nonlinear programming

Keywords: minimizing convex functions; nonexpansive mappings; intersection of the fixed point sets; cyclic hybrid steepest descent algorithm

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