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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

$x\in \bigcap _{i=1}^{p}{C}_{i}\phantom{\rule{1.em}{0ex}}\text{with}\phantom{\rule{1.em}{0ex}}{T}_{j}x\in {Q}_{j},\phantom{\rule{1.em}{0ex}}j\in \left\{1,\cdots ,r\right\}·$

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

$f\left(x\right)=\frac{1}{2}\sum _{i=1}^{p}{\alpha }_{i}|x-{P}_{{C}_{i}}{\left(x\right)|}^{2}+\frac{1}{2}\sum _{j=1}^{r}{\beta }_{j}{|{T}_{j}x-{P}_{{Q}_{j}}\left({T}_{j}x\right)|}^{2}$

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 Methods of nonlinear programming type in calculus of variations 47H09 Mappings defined by “shrinking” properties 47N10 Applications of operator theory in optimization, convex analysis, programming, economics