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**A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets.**
*(English)*
Zbl 1164.47399

Summary: The method of projections onto convex sets, to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such a corridor consists in taking a big step at different moments during the iteration, because in that way the monotonous behavior that is responsible for the slow convergence may be interrupted. In this paper, we present a technique that may introduce interruption of the monotony for a sequential algorithm, but at the same time guarantees convergence of the constructed sequence to a point in the intersection of the sets. Concerning convergence speed, we compare experimentally the behavior of the new algorithm with that of an existing monotonous algorithm.

### MSC:

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

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

40A99 | Convergence and divergence of infinite limiting processes |

49M30 | Other numerical methods in calculus of variations (MSC2010) |

52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |

90C25 | Convex programming |

### References:

[1] | H. Bauschke and J. Borwein On projection algorithms for solving convex feasibility problems. Siam Review 38 (1996), 367–426. · Zbl 0865.47039 |

[2] | D. Butnariu and Y. Censor: On the behaviour of a block-iterative projection method for solving convex feasibility problems. Intern. J. Computer Math. 34 (1990), 79–94. · Zbl 0708.90064 |

[3] | Y. Censor and S. A. Zenios: Parallel optimization. Theory, algorithms, and applications, Oxford University Press, Inc., New York, 1997. |

[4] | G. Crombez: Viewing parallel projection methods as sequential ones in convex feasibility problems. Trans. Amer. Math. Soc. 347 (1995), 2575–2583. · Zbl 0846.46010 |

[5] | G. Crombez: Improving the speed of convergence in the method of projections onto convex sets. Publicationes Mathematicae Debrecen 58 (2001), 29–48. · Zbl 0973.65001 |

[6] | F. Deutsch: The method of alternating orthogonal projections. In: ”Approximation theory, spline functions and applications”, Kluwer Academic Publishers, 1992, pp. 105–121. · Zbl 0751.41031 |

[7] | J. Dye and S. Reich: Random products of nonexpansive mappings. In: ”Optimization and Nonlinear Analysis”, Pitman Research Notes in Mathematics Series, Vol. 244, Longman, Harlow, 1992, pp. 106–118. · Zbl 0815.47067 |

[8] | W. Gearhart and M. Koshy: Acceleration schemes for the method of alternating projections. J. Comp. Appl. Math. 26 (1989), 235–249. · Zbl 0688.65040 |

[9] | 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. · Zbl 0199.51002 |

[10] | M. Hanke and W. Niethammer: On the acceleration of Kaczmarz’s method for inconsistent linear systems. Linear Algebra Appl. 130 (1990), 83–98. · Zbl 0708.65033 |

[11] | D. Schott: Iterative solution of convex problems by Fejér-monotone methods. Numer. Funct. Anal. and Optimiz. 16 (1995), 1323–1357. · Zbl 0853.65055 |

[12] | H. Stark and Y. Yang: Vector space projections. J. Wiley & Sons, Inc., New York, 1998. · Zbl 0903.65001 |

[13] | L. Vandenberghe and S. Boyd: Semidefinite programming. Siam Review 38 (1996), 49–95. · Zbl 0845.65023 |

[14] | Y. Yang, N. Galatsanos and A. Katsaggelos: Projection-based spatially adaptive reconstruction of block-transform compressed images. IEEE Trans. Image Processing 4 (1995), 896–908. |

[15] | D. C. Youla: Mathematical theory of image restoration by the method of convex projections. In: H. Stark (editor), ”Image recovery: theory and applications”, Academic Press, New York, 1987, pp. 29–77. |

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