@article{809c35cc47624195a7db8eb7203d8900,
title = "A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections",
abstract = "The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the ε-under-relaxed cyclic projection method converge when ε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in R3 for which the ε-under-relaxed cycles do not converge.",
keywords = "Cyclic projections, De Pierro conjecture, under-relaxed projections",
author = "Roberto Cominetti and Vera Roshchina and Andrew Williamson",
note = "Funding Information: Roberto Cominetti was partially support from FONDECYT [1171501] and N{\'u}cleo Milenio ICM/FIC RC130003 {\textquoteleft}Informaci{\'o}n y Coordinaci{\'o}n en Redes{\textquoteright}. Vera Roshchina{\textquoteright}s research was supported by the Australian Research Council [grant number DE150100240] and Enabling Capability Platform Information & Systems (Engineering) of RMIT University. Funding Information: Roberto Cominetti was partially support from FONDECYT [1171501] and N?cleo Milenio ICM/FIC RC130003 ?Informaci?n y Coordinaci?n en Redes?. Vera Roshchina?s research was supported by the Australian Research Council [grant number DE150100240] and Enabling Capability Platform Information & Systems (Engineering) of RMIT University. Publisher Copyright: {\textcopyright} 2018, {\textcopyright} 2018 Informa UK Limited, trading as Taylor & Francis Group.",
year = "2019",
month = jan,
day = "2",
doi = "10.1080/02331934.2018.1474471",
language = "English",
volume = "68",
pages = "3--12",
journal = "Optimization",
issn = "0233-1934",
publisher = "Taylor and Francis Ltd.",
number = "1",
}