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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 \[ x\in\bigcap^p_{i=1} C_i\quad\text{with}\quad T_jx\in Q_j,\quad j\in \{1,\dots, r\}. \] Here, \(C_i\in H_1\) and \(Q_j\in H_2\) are closed convex subsets of the Hilbert spaces \(H_1\) and \(H_2\), respectively, and \(T_j:H_1\to H_2\) are bounded linear operators. The method bases on the minimization of the function \[ f(x)= {1\over 2} \sum^p_{i=1} \alpha_i|x- P_{C_i}(x)|^2+ {1\over 2} \sum^r_{j=1}\beta_j|T_j x- P_{Q_j}(T_j x)|^2 \] with suitable positive parameters \(\alpha_i\) and \(\beta_j\) (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.

MSC:

90C25 Convex programming
49M37 Numerical methods based on nonlinear programming
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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