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Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. (English) Zbl 0913.47048
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 $u^*$.

47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
90C25Convex programming
65K05Mathematical programming (numerical methods)
65K10Optimization techniques (numerical methods)
90C30Nonlinear programming
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