## Iterative oblique projection onto convex sets and the split feasibility problem.(English)Zbl 0996.65048

Let $$C$$ and $$Q$$ be nonempty closed convex sets in $${\mathbb R}^N$$ and $${\mathbb R}^M$$, respectively, and $$A$$ an $$M$$ by $$N$$ real matrix. The split feasibility problem (SFP) is to find $$x\in C$$ with $$Ax\in Q$$, if such $$x$$ exits. The author presents an iterative method for solving the SFP, called the $$CQ$$ algorithm, which has the following iterative step: $$x^{k+1}=P_C(x^k+\gamma A^T(P_Q-I)Ax^k)$$, where $$\gamma\in(0,2/L)$$ with $$L$$ the largest eigenvalue of $$A^TA$$ and $$P_C$$ and $$P_Q$$ denote the orthogonal projections onto $$C$$ and $$Q$$, respectively.
It is shown that the $$CQ$$ algorithm converges to a solution of the SFP, or, more generally, to a minimizer of $$\|P_CAc-Ac\|$$ over $$c\in C$$, whenever such $$c$$ exists. The $$CQ$$ algorithm involves only the easily calculated projections $$P_C$$ and $$P_Q$$, but no matrix inverses. If $$A$$ is normalized so that each row has length one, then $$L$$ does not exceed the maximum number of nonzero entries in any column of $$A$$. This estimation of $$L$$ accelerates the convergence by permitting the relaxation parameter $$\gamma$$ to take on larger values.
The $$CQ$$ algorithm can be extended to a block-iterative version which may provide further acceleration. One promising application of the $$CQ$$ algorithm is dynamic emission tomographic image reconstruction.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 65F10 Iterative numerical methods for linear systems
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