The composition of projections onto closed convex sets in Hilbert space is asymptotically regular

Author:
Heinz H. Bauschke

Journal:
Proc. Amer. Math. Soc. **131** (2003), 141-146

MSC (2000):
Primary 47H05, 47H09, 90C25

DOI:
https://doi.org/10.1090/S0002-9939-02-06528-0

Published electronically:
May 9, 2002

MathSciNet review:
1929033

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Abstract: The composition of finitely many projections onto closed convex sets in Hilbert space arises naturally in the area of projection algorithms. We show that this composition is asymptotically regular, thus proving the so-called ``zero displacement conjecture'' of Bauschke, Borwein and Lewis. The proof relies on a rich mix of results from monotone operator theory, fixed point theory, and convex analysis.

**1.**H. H. Bauschke.*Projection algorithms and monotone operators*.

Ph.D. thesis, Simon Fraser University, 1996.

Available at`http://www.cecm.sfu.ca/preprints/1996pp.html`.**2.**H. H. Bauschke and J. M. Borwein.

On projection algorithms for solving convex feasibility problems.*SIAM Review*, 38:367-426, 1996. MR**98f:90045****3.**H. H. Bauschke and J. M. Borwein.

Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators.*Pacific Journal of Mathematics*, 189(1):1-20, 1999. MR**2001j:47059****4.**H. H. Bauschke, J. M. Borwein, and A. S. Lewis.

The method of cyclic projections for closed convex sets in Hilbert space.

In*Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995)*, pages 1-38. Amer. Math. Soc., Providence, RI, 1997. MR**98c:49069****5.**L. M. Bregman.

The method of successive projection for finding a common point of convex sets.*Soviet Math. Dokl.*, 6:688-692, 1965.**6.**H. Brézis and A. Haraux.

Image d'une somme d'opérateurs monotones et applications.*Israel Journal of Mathematics*, 23(2):165-186, 1976. MR**53:3803****7.**F. E. Browder and W. V. Petryshin.

The solution by iteration of nonlinear functional equations in Banach spaces.*Bulletin of the American Mathematical Society*, 72:571-575, 1966. MR**32:8155b****8.**R. E. Bruck and S. Reich.

Nonexpansive projections and resolvents of accretive operators in Banach spaces.*Houston Journal of Mathematics*, 3(4):459-470, 1977. MR**57:10507****9.**A. R. De Pierro.

From parallel to sequential projection methods and vice versa in convex feasibility: results and conjectures.

In D. Butnariu, Y. Censor, and S. Reich (editors)*Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (Haifa, 2000)*, pages 187-202, Elsevier 2001.**10.**K. Goebel and W. A. Kirk.*Topics in metric fixed point theory*.

Cambridge University Press, 1990. MR**92c:47070****11.**R. R. Phelps.*Convex functions, monotone operators and differentiability*.

Springer-Verlag, Berlin, second edition, 1993. MR**94f:46055****12.**R. R. Phelps and S. Simons.

Unbounded Linear Monotone Operators on Nonreflexive Banach Spaces.*Journal of Convex Analysis*, 5(2):303-328, 1998. MR**99k:47003****13.**Stephen Simons.*Minimax and monotonicity*.

Springer-Verlag, Berlin, 1998. MR**2001h:49002**

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Additional Information

**Heinz H. Bauschke**

Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

Email:
hbauschk@uoguelph.ca

DOI:
https://doi.org/10.1090/S0002-9939-02-06528-0

Received by editor(s):
February 20, 2001

Received by editor(s) in revised form:
August 13, 2001

Published electronically:
May 9, 2002

Additional Notes:
The author’s research was supported by NSERC

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2002
American Mathematical Society