Projections onto convex sets

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 convex sets. It is a very simple algorithm and has been rediscovered many times.[1] The simplest case, when the sets are affine spaces, was analyzed by John von Neumann.[2][3] 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.[4][5] There are now extensions that consider cases when there are more than two sets, or when the sets are not convex,[6] 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 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 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.[7]

  1. ^ Bauschke, H.H.; Borwein, J.M. (1996). "On projection algorithms for solving convex feasibility problems". SIAM Review. 38 (3): 367–426. CiteSeerX 10.1.1.49.4940. doi:10.1137/S0036144593251710.
  2. ^ J. von Neumann,Neumann, John Von (1949). "On rings of operators. Reduction theory". Ann. of Math. 50 (2): 401–485. doi:10.2307/1969463. JSTOR 1969463. (a reprint of lecture notes first distributed in 1933)
  3. ^ J. von Neumann. Functional Operators, volume II. Princeton University Press, Princeton, NJ, 1950. Reprint of mimeographed lecture notes first distributed in 1933.
  4. ^ Gubin, L.G.; Polyak, B.T.; Raik, E.V. (1967). "The method of projections for finding the common point of convex sets". U.S.S.R. Computational Mathematics and Mathematical Physics. 7 (6): 1–24. doi:10.1016/0041-5553(67)90113-9.
  5. ^ Bauschke, H.H.; Borwein, J.M. (1993). "On the convergence of von Neumann's alternating projection algorithm for two sets". Set-Valued Analysis. 1 (2): 185–212. doi:10.1007/bf01027691. S2CID 121602545.
  6. ^ Lewis, Adrian S.; Malick, Jérôme (2008). "Alternating Projections on Manifolds". Mathematics of Operations Research. 33: 216–234. CiteSeerX 10.1.1.416.6182. doi:10.1287/moor.1070.0291.
  7. ^ Combettes, P. L. (1993). "The foundations of set theoretic estimation" (PDF). Proceedings of the IEEE. 81 (2): 182–208. doi:10.1109/5.214546. Archived from the original (PDF) on 2015-06-14. Retrieved 2012-10-09.