Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings

@article{Deutsch1998MinimizingCC,
  title={Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings},
  author={Frank Deutsch and Isao Yamada},
  journal={Numerical Functional Analysis and Optimization},
  year={1998},
  volume={19},
  pages={33-56}
}
Let T i (i = 1,2,...,N) be nonexpansive mappings on a Hilbert space H, and let ⊖: H → R∪{∞} be a function which has a uniformly strongly positive and uniformly bounded second (Frechet) derivative over the convex hull of T i (H) for some i. We first prove that ⊖ 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 we prove that it converges to u*. This generalizes some recent results… 
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