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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 \({\mathcal H}\), and let \(\Theta:{\mathcal H}\to \mathbb{R}\cup \{\infty\}\) be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of \(T_i({\mathcal H})\) 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 \(u^*\).

MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
90C25 Convex programming
65K05 Numerical mathematical programming methods
65K10 Numerical optimization and variational techniques
90C30 Nonlinear programming
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