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Zbl 1171.90009
Masad, Eyal; Reich, Simeon
A note on the multiple-set split convex feasibility problem in Hilbert space.
(English)
[J] J. Nonlinear Convex Anal. 8, No. 3, 367-371 (2007). ISSN 1345-4773; ISSN 1880-5221/e

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).\par 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.
[Jörg Thierfelder (Ilmenau)]
MSC 2000:
*90C25 Convex programming
49M37 Methods of nonlinear programming type
47H09 Mappings defined by "shrinking" properties
47N10 Appl. of operator theory in optimization, math. programming, etc.

Keywords: convex feasibility problem; gradient method; nearest point projection; nonexpansive mapping; averaged mapping; strong convergence

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