Projections onto convex sets: Difference between revisions

Revision as of 09:28, 12 June 2015

Revision as of 09:29, 12 June 2015

Line 3:

<ref>L.G. Gubin, B.T. Polyak, and E.V. Raik. The method of projections for finding the common point of convex sets. U.S.S.R. Computational Mathematics and Mathematical Physics, 7:1–24, 1967.</ref>

<ref>H.H. Bauschke and J.M. Borwein. On the convergence of von Neumann's alternating projection algorithm for two sets. Set-Valued Analysis, 1:185–212, 1993.</ref>

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 DOI| 10.1287/moor.1070.0291}}</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_Projection (linear_algebralinear algebra)#Orthogonal_projectionsOrthogonal 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">P. L. Combettes, "The foundations of set theoretic estimation," Proceedings of the IEEE, vol. 81, no. 2, pp. 182–208, February 1993. [http://www.ann.jussieu.fr/~plc/proc.pdf PDF]</ref>

== Algorithm ==