Projections onto convex sets

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 in fact to the orthogonal projection onto the intersection of the initial iterate. 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 one set, or when the sets are not convex^[6]. 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.

Algorithm

The POCS algorithm solves the following problem:

${\text{find}}\;x\in {\mathcal {R}}^{n}\quad {\text{such that}}\;x\in C\cap D$

where C and D are closed convex sets.

To use the POCS algorithm, one must know how to project onto the sets C and D separately. The algorithm starts with an arbitrary value for $x_{0}$ and then generate the sequence

$x_{k+1}={\mathcal {P}}_{C}\left({\mathcal {P}}_{D}(x_{k})\right)$ .

If the intersection of C and D is non-empty, then the sequence generated by the algorithm will converge to some point in this intersection.

Unlike Dykstra's projection algorithm, the solution need not be a projection onto the intersection C and D.

Related algorithms

The method of averaged projections is quite similar. For the case of two closed convex sets C and D, it proceeds by

$x_{k+1}={\frac {1}{2}}({\mathcal {P}}_{C}(x_{k})+{\mathcal {P}}_{D}(x_{k}))$

It has long been known to converge globally.^[7] Furthermore, the method is easy to generalize to more than two sets; some convergence results for this case are in ^[8].

References

^ H.H. Bauschke and J.M. Borwein. On projection algorithms for solving convex feasibility problems. SIAM Review, 38(3):367–426, 1996.
^ J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949) 401–485 (a reprint of lecture notes first distributed in 1933).
^ J. von Neumann. Functional Operators, volume II. Princeton University Press, Princeton, NJ, 1950. Reprint of mimeographed lecture notes first distributed in 1933.
^ 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.
^ H.H. Bauschke and J.M. Borwein. On the convergence of von Neu- mann’s alternating projection algorithm for two sets. Set-Valued Anal- ysis, 1:185–212, 1993.
^ Template:Cite DOI
^ A. Auslender. Methodes Numeriques pour la Resolution des Problems d’Optimisation avec Constraintes. PhD thesis, Faculte des Sciences, Grenoble, 1969
^ Local convergence for alternating and averaged nonconvex projections. A Lewis, R Luke, J Malick, 2007. arXiv

Algorithm

Related algorithms

References

Further reading