Viewing parallel projection methods as sequential ones in convex feasibility problems.

*(English)*Zbl 0846.46010Summary: We show that the parallel projection method with variable weights and one variable relaxation coefficient for obtaining a point in the intersection of a finite number of closed convex sets in a given Hilbert space may be interpreted as a semi-alternating sequential projection method in a suitably newly constructed Hilbert space. As such, convergence results for the parallel projection method may be derived from those which may be constructed in the semi-alternating sequential case.

##### MSC:

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

68U10 | Computing methodologies for image processing |

##### Keywords:

convex feasibility problem; block-iterative projection method; parallel projection method with variable weights; one variable relaxation coefficient; intersection of a finite number of closed convex sets in a given Hilbert space; semi-alternating sequential projection method
PDF
BibTeX
XML
Cite

\textit{G. Crombez}, Trans. Am. Math. Soc. 347, No. 7, 2575--2583 (1995; Zbl 0846.46010)

Full Text:
DOI

**OpenURL**

##### References:

[1] | D. Butnariu and Y. Censor, On the behavior of a block-iterative projection method for solving convex feasibility problems, Internat. J. Computer Math. 43 (1990), 79-94. · Zbl 0708.90064 |

[2] | G. Crombez, Weak and norm convergence of a parallel projection method in Hilbert spaces, Appl. Math. Comput. 56 (1993), no. 1, 35 – 48. · Zbl 0783.65049 |

[3] | -, A parallel projection method based on sequential most remote set in convex feasibility problems (submitted for publication). · Zbl 0840.65039 |

[4] | A. R. De Pierro, An extended decomposition through formalization in product spaces, preprint. |

[5] | A. R. De Pierro and A. N. Iussem, A parallel projection method of finding a common point of a family of convex sets, Pesquisa Operacional 5 (1985), 1-20. |

[6] | L. G. Gubin, B. T. Polyak, and E. V. Raik, The method of projections for finding the common point of convex sets, USSR Comput. Math. and Math. Phys. 7 (1967), 1-24. |

[7] | N. Ottavy, Strong convergence of projection-like methods in Hilbert spaces, J. Optim. Theory Appl. 56 (1988), no. 3, 433 – 461. · Zbl 0621.49019 |

[8] | G. Pierra, Decomposition through formalization in a product space, Math. Programming 28 (1984), no. 1, 96 – 115. · Zbl 0523.49022 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.