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Strong convergence of expected-projection methods in Hilbert spaces. (English) Zbl 0834.65041
The paper deals with stochastic convex feasibility problems (SCFPs) which particularly reduce to convex feasibility problems (CFPs), a well-known and useful tool in applied science. The paper shows that the so-called projection methods, the most commonly used tools to solve CFPs also apply to solve infinite SCFPs.
To this end in section 2 expected-projection operators and in section 3 relaxed expected-projection operators are defined and studied. Next, in section 4 a strong convergence criterion for expected-projection methods (EPM) is stated and proved. However, the sufficient conditions for strong convergence of EPM are implicit conditions, except for (C5).
So, section 5 is devoted to obtain explicit strong convergence criteria for EPMs, in order to be able to effectively solve a wide class of SCFPs by expected-projection methods. Finally, section 6 gives applications and examples of the above results to: stochastic systems of convex inequalities, best approximation problems, and computational tomography.

MSC:
65J05 General theory of numerical analysis in abstract spaces
65K05 Numerical mathematical programming methods
47H40 Random nonlinear operators
90C15 Stochastic programming
41A50 Best approximation, Chebyshev systems
92C55 Biomedical imaging and signal processing
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