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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,,N) be nonexpansive mappings on a Hilbert space , and let Θ:{} be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of T i () for some i. The authors prove that Θ 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:
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