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A note on the multiple-set split convex feasibility problem in Hilbert space. (English) Zbl 1171.90009

The authors provide a gradient method for the (constrained) multiple-set split convex feasibility problem, finding a point

x i=1 p C i withT j xQ j ,j{1,,r}·

Here, C i H 1 and Q j H 2 are closed convex subsets of the Hilbert spaces H 1 and H 2 , respectively, and T j :H 1 H 2 are bounded linear operators. The method bases on the minimization of the function

f(x)=1 2 i=1 p α i |x-P C i (x)| 2 +1 2 j=1 r β j |T j x-P Q j (T j x)| 2

with suitable positive parameters α i and β j (where P denotes the projection operator).

It is shown that the iteration sequence generated by the method converges weakly to a solution of the problem. Assuming additional properties for the sets C i and Q j , even strong convergence can be proved.

90C25Convex programming
49M37Methods of nonlinear programming type in calculus of variations
47H09Mappings defined by “shrinking” properties
47N10Applications of operator theory in optimization, convex analysis, programming, economics