Bauschke, Heinz H. and Borwein, Jonathan M. and Lewis, Adrian (1994) *On the method of cyclic projections for convex sets in Hilbert space.* [Preprint]

## Abstract

The method of cyclic projections is a powerful tool for solving convex feasibility problems in Hilbert space. Although in many applications, in particular in the field of image reconstruction (electron microscopy, computed tomography), the convex constraint sets do not necessarily intersect, the method of cyclic projections is still employed. Results on the behaviour of the algorithm for this general case are improved, unified, and reviewed. The analysis relies on key concepts from convex analysis and the theory of nonexpansive mappings. The notion of the angle of a tuple of subspaces is introduced. New linear convergence results follow for the case when the constraint sets are closed subspaces whose orthogonal complements have a closed sum; this holds, in particular, for hyperplanes or in Euclidean space.

Item Type: | Preprint |

Additional Information: | pubdom FALSE |

Uncontrolled Keywords: | angle, computerized tomography, convex feasibility problem, convex polyhedron, convex set, Fejer monotone sequence, Hilbert space, image reconstruction, Kaczmarz's method, method of cyclic projections, nearest point mapping, nonexpansive mapping, projection, sum of subspaces, von Neumann's alternating projection algorithm |

Subjects: | 92-xx Biology and other natural sciences, behavioral sciences > 92Cxx Physiological, cellular and medical topics 49-xx Calculus of variations and optimal control; optimization > 49Mxx Methods of successive approximations 65-xx Numerical analysis > 65Jxx Numerical analysis in abstract spaces 65-xx Numerical analysis > 65Kxx Mathematical programming, optimization and variational techniques 52-xx Convex and discrete geometry > 52Axx General convexity 65-xx Numerical analysis > 65Fxx Numerical linear algebra 46-xx Functional analysis > 46Cxx Inner product spaces and their generalizations, Hilbert spaces 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming 46-xx Functional analysis > 46Nxx Miscellaneous applications of functional analysis 26-xx Real functions > 26Bxx Functions of several variables 65-xx Numerical analysis 47-xx Operator theory > 47Hxx Nonlinear operators and their properties 41-xx Approximations and expansions > 41Axx Approximations and expansions 47-xx Operator theory > 47Nxx Miscellaneous applications of operator theory |

Faculty: | UNSPECIFIED |

Depositing User: | *Users 1 not found.* |

Date Deposited: | 16 Nov 2003 |

Last Modified: | 21 Apr 2010 11:13 |

URI: | https://docserver.carma.newcastle.edu.au/id/eprint/77 |
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