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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 name="SIAMreview" /> The simplest case, when the sets are [[affine spaces]], was analyzed by [[John von Neumann]].<ref>{{cite journal | year = 1949 | title = On rings of operators. Reduction theory | doi = 10.2307/1969463 | journal = Ann. of Math. | volume = 50 | issue = 2| pages = 401–485 }} (a reprint of lecture notes first distributed in 1933)</ref>
<ref>J. von Neumann. Functional Operators, volume II. Princeton University Press, Princeton, NJ, 1950. Reprint of mimeographed lecture notes first distributed in 1933.</ref> The case when the sets are affine spaces is special, since the iterates not only converge to a point in the intersection (assuming the intersection is non-empty) but
<ref>{{cite journal | last1 = Gubin | first1 = L.G. | last2 = Polyak | first2 = B.T. | last3 = Raik | first3 = E.V. | year = 1967 | title = The method of projections for finding the common point of convex sets | url = | journal = U.S.S.R. Computational Mathematics and Mathematical Physics | volume = 7 | issue = | pages = 1–24 | doi=10.1016/0041-5553(67)90113-9}}</ref>
<ref>{{cite journal | last1 = Bauschke | first1 = H.H. | last2 = Borwein | first2 = J.M. | year = 1993 | title = On the convergence of von Neumann's alternating projection algorithm for two sets | url = | journal = Set-Valued Analysis | volume = 1 | issue = | pages = 185–212 | doi=10.1007/bf01027691}}</ref>
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