Projections onto convex sets

<p>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 <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Convex_set/vdAuJRJl">convex</a> sets. It is a very simple algorithm and has been rediscovered many times. The simplest case, when the sets are <a href="/facts/Affine_spaces/tlvI3oX1">affine spaces</a>, was analyzed by <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a>. 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 <a href="/facts/Rate_of_convergence/JVGuzPoS">rate of convergence</a> of the iterates is linear.
There are now extensions that consider cases when there are more than two sets, or when the sets are not <a href="/facts/Convex_set/vdAuJRJl">convex</a>, or that give faster convergence rates. Analysis of POCS and related methods attempt to show that the algorithm converges (and if so, find the <a href="/facts/Rate_of_convergence/JVGuzPoS">rate of convergence</a>), and whether it converges to the <a href="/facts/Projection_(linear_algebra)/sElXGkxD">projection</a> 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 <a href="/facts/Dykstra%2527s_projection_algorithm/jhYGXoeK">Dykstra's projection algorithm</a>.  See the references in the 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.
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Projections onto convex sets open-in-new

Projections onto convex sets