## 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