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

Bauschke, Heinz

doi:10.1090/S0002-9939-02-06528-0

The composition of projections onto closed convex sets in Hilbert space is asymptotically regular
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by Heinz H. Bauschke PDF

Proc. Amer. Math. Soc. 131 (2003), 141-146 Request permission

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.

References

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.
Heinz H. Bauschke and Jonathan M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38 (1996), no. 3, 367–426. MR 1409591, DOI 10.1137/S0036144593251710
Heinz H. Bauschke and Jonathan M. Borwein, Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators, Pacific J. Math. 189 (1999), no. 1, 1–20. MR 1687802, DOI 10.2140/pjm.1999.189.1
Heinz H. Bauschke, Jonathan M. Borwein, and Adrian S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space, Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995) Contemp. Math., vol. 204, Amer. Math. Soc., Providence, RI, 1997, pp. 1–38. MR 1442992, DOI 10.1090/conm/204/02620
L. M. Bregman. The method of successive projection for finding a common point of convex sets. Soviet Math. Dokl., 6:688–692, 1965.
Haïm Brezis and Alain Haraux, Image d’une somme d’opérateurs monotones et applications, Israel J. Math. 23 (1976), no. 2, 165–186. MR 399965, DOI 10.1007/BF02756796
F. E. Browder and W. V. Petryshyn, The solution by iteration of linear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 566–570. MR 190744, DOI 10.1090/S0002-9904-1966-11543-4
Ronald E. Bruck and Simeon Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3 (1977), no. 4, 459–470. MR 470761
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.
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
Robert R. Phelps, Convex functions, monotone operators and differentiability, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR 1238715
R. R. Phelps and S. Simons, Unbounded linear monotone operators on nonreflexive Banach spaces, J. Convex Anal. 5 (1998), no. 2, 303–328. MR 1670360
Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. MR 1723737, DOI 10.1007/BFb0093633