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Abstract

The split feasibility problem (SFP) is important due to its occurrence in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy. Mathematically, it can be formulated as finding a point x∗ such that x∗ ∈ C and Ax∗ ∈ Q, where A is a bounded linear operator, C and Q are subsets of two Hilbert spaces H₁ and H₂ respectively. One particular algorithm for solving this problem is the CQ algorithm. In this thesis, previous work on CQ algorithm is presented and a new proof of convergence of the relaxed CQ algorithm is given. The CQ algorithm is shown to be a special case of the subgradient projection algorithm. The SFP is extended into two nonconvex cases. The first one is on S-subdifferentiable functions, and the other one is on prox-regular functions. The subgradient projection algorithm and CQ algorithm are proved to converge to a solution of the first and second case respectively.