## The relaxed CQ algorithm solving the split feasibility problem.(English)Zbl 1066.65047

Let $$C$$ and $$Q$$ be nonempty closed convex sets in $${\mathbb R}^N$$ and $${\mathbb R}^M$$, 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 $$C=\{x\in {\mathbb R}^N\,| \, c(x)\leq 0\}$$ and $$Q=\{y\in {\mathbb R}^M\,| \, q(y)\leq 0\}$$, where $$c$$ and $$q$$ are convex functions, by the larger sets $$C_k=\{x\in {\mathbb R}^N\,| \, c(x^k)+\langle \xi^k,x-x^k\rangle \leq 0\}$$ and $$Q_k=\{y\in {\mathbb R}^M\,| \, q(Ax^k)+\langle \eta^k,y-Ax^k\rangle \leq 0\}$$ 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 (MSC2010) 65F10 Iterative numerical methods for linear systems
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