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Block-iterative surrogate projection methods for convex feasibility problems. (English) Zbl 0821.65037
The author gives projection methods for solving the set intersection problem (SIP): find, if possible, any point in \(\bigcap^ m_{i=1} C_ i\), and the convex feasibility problem (CFP): find, if possible, any point \(x\), such that \(f_ i(x) \leq 0\), \(i = 1, \dots, m\), where each \(C_ i\) is a closed convex subset of \(\mathbb{R}^ n\) and each function \(f_ i \in \mathbb{R}^ n \to \mathbb{R}\) is convex.
In this paper it is shown that many algorithms for SIP or CFP fit a geometric framework. In this geometric framework, algorithms for CFP are treated as methods for SIP that approximate the sets \(C_ i\) via half spaces. In section 2 of the paper an abstract method is described that employs approximating half spaces under general conditions which simplify its convergence analysis. In the further section examples of surrogate cuts, a surrogate projection method and block-iterative methods are discussed.

MSC:
65K05 Numerical mathematical programming methods
90C25 Convex programming
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