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{{confusing|date=April 2018}}
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 = Bauschke | first1 = H.H. | last2 = Borwein | first2 = J.M. | year = 1996 | title = On projection algorithms for solving convex feasibility problems | doi = 10.1137/S0036144593251710 | journal = SIAM Review | volume = 38 | issue = 3| pages = 367–426 | citeseerx = 10.1.1.49.4940 }}</ref> The simplest case, when the sets are [[affine spaces]], was analyzed by [[John von Neumann]].<ref>J. von Neumann,{{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 | jstor = 1969463 | last1 = Neumann | first1 = John Von }} (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 to the orthogonal projection of the point onto the intersection. For general closed convex sets, the limit point need not be the projection. Classical work on the case of two closed convex sets shows that the [[rate of convergence]] of the iterates is linear.<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
There are now extensions that consider cases when there are more than two sets, or when the sets are not [[convex set|convex]],<ref>{{Cite journal
▲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 = Bauschke | first1 = H.H. | last2 = Borwein | first2 = J.M. | year = 1996 | title = On projection algorithms for solving convex feasibility problems | doi = 10.1137/S0036144593251710 | journal = SIAM Review | volume = 38 | issue = 3| pages = 367–426 }}</ref> The simplest case, when the sets are [[affine spaces]], was analyzed by [[John von Neumann]].<ref>J. von Neumann,{{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 to the orthogonal projection of the point onto the intersection. For general closed convex sets, the limit point need not be the projection. Classical work on the case of two closed convex sets shows that the [[rate of convergence]] of the iterates is linear.<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>
| last1=Lewis | first1=Adrian S.
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>▼
| last2=Malick | first2=Jérôme
| doi=10.1287/moor.1070.0291
| title=Alternating Projections on Manifolds
| journal=[[Mathematics of Operations Research]]
| volume=33
| pages=216–234
| date=2008
▲
== Algorithm ==
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The POCS algorithm solves the following problem:
: <math> \text{find} \; x \in \
where ''C'' and ''D'' are [[closed set|closed]] [[convex set]]s.
To use the POCS algorithm, one must know how to project onto the sets ''C'' and ''D'' separately, via the projections <math>\mathcal{P}_i</math>.
The algorithm starts with an arbitrary value for <math>x_0</math> and then generates the sequence
Line 30 ⟶ 39:
It has long been known to converge globally.<ref>A. Auslender. Methodes Numeriques pour la Resolution des Problems
d’Optimisation avec Constraintes. PhD thesis, Faculte des Sciences, Grenoble, 1969</ref> Furthermore, the method is easy to generalize to more than two sets; some convergence results for this case are in.<ref>
| | | | title=Local convergence for alternating and averaged nonconvex projections
| journal=Foundations of Computational Mathematics
| volume=9
| issue=4
| pages=485–513
| date=2009
| arxiv=0709.0109
| doi=10.1007/s10208-008-9036-y}}</ref>
The ''averaged'' projections method can be reformulated as ''alternating'' projections method using a standard trick. Consider the set
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: <math> E = \{ (x,y) : x \in C, \; y \in D \}</math>
which is defined in the [[Tensor product|product space]] <math> \
Then define another set, also in the product space:
: <math> F = \{ (x,y) : x \in \
Thus finding <math> C \cap D </math> is equivalent to finding <math> E \cap F</math>.
Line 53 ⟶ 73:
== Further reading ==
* Book from 2011: [
[[Category:Convex geometry]]
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