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The relaxed CQ algorithm solving the split feasibility problem. (English) Zbl 1066.65047
Let $C$ and $Q$ be nonempty closed convex sets in ${\Bbb R}^N$ and ${\Bbb R}^M$, respectively, and $A$ an $M\times N$ real matrix. The split feasibility problem (SFP) is to find $x\in C$ with $Ax\in Q$, if such $x$ exits. The author presents the following relaxed CQ algorithm: $x^{k+1}=P_{C_k}(x^k+\gamma A^T(P_{Q_k}-I)Ax^k)$ for solving the SFP by replacing the sets $C=\{x\in {\Bbb R}^N\,\vert \, c(x)\leq 0\}$ and $Q=\{y\in {\Bbb R}^M\,\vert \, q(y)\leq 0\}$, where $c$ and $q$ are convex functions, by the larger sets $C_k=\{x\in {\Bbb R}^N\,\vert \, c(x^k)+\langle \xi^k,x-x^k\rangle \leq 0\}$ and $Q_k=\{y\in {\Bbb R}^M\,\vert \, q(Ax^k)+\langle \eta^k,y-Ax^k\rangle \leq 0\}$ where the subgradients are $\xi^k\in\partial c(x^k)$ and $\eta^k\in\partial q(Ax^k)$. Here $P_{C_k}$ and $P_{Q_k}$ are projections onto $C_k$ and $Q_k$, and $\gamma\in(0,2/L)$ where $L$ denotes the largest eigenvalue of $A^TA$. The convergence of the algorithm is established and another algorithm for SFP is given.

##### MSC:
 65F30 Other matrix algorithms 65F10 Iterative methods for linear systems
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