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In mathematics, '''projections onto convex sets''' ('''POCS'''), sometimes known as the '''alternating projection''' method, is a method to find a point in the intersection of two [[closed set|closed]] [[convex set|convex]] sets. It is a very simple algorithm and has been rediscovered many times.<ref>{{cite journal | last1
There are now extensions that consider cases when there are more than one set, or when the sets are not [[convex set|convex]],<ref>{{Cite journal | last1 = Lewis | first1 = A. S. | last2 = Malick | first2 = J. | doi = 10.1287/moor.1070.0291 | title = Alternating Projections on Manifolds | journal = Mathematics of Operations Research | volume = 33 | pages = 216–234 | year = 2008 | pmid = | pmc = }}</ref> or that give faster convergence rates. Analysis of POCS and related methods attempt to show that the algorithm converges (and if so, find the [[rate of convergence]]), and whether it converges to the [[Projection (linear algebra)#Orthogonal projections|projection]] of the original point. These questions are largely known for simple cases, but a topic of active research for the extensions. There are also variants of the algorithm, such as [[Dykstra's projection algorithm]]. See the references in the [[#Further reading|further reading]] section for an overview of the variants, extensions and applications of the POCS method; a good historical background can be found in section III of.<ref name="PLC">{{cite journal | last1 = Combettes | first1 = P. L. | year = 1993 | title = The foundations of set theoretic estimation | url = http://www.ann.jussieu.fr/~plc/proc.pdf | format = PDF | journal = Proceedings of the IEEE | volume = 81 | issue = 2| pages = 182–208 | doi=10.1109/5.214546}}</ref>
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== Further reading ==
* Book from 2011: [http://www.ec-securehost.com/SIAM/FA08.html Alternating Projection Methods] by René Escalante and Marcos Raydan (2011), published by SIAM.
[[Category:Convex geometry]]
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