Strong convergence of alternating projections. (English) Zbl 07536869
Summary: In this paper, we provide a necessary and sufficient condition under which the method of alternating projections on Hadamard spaces converges strongly. This result is new even in the context of Hilbert spaces. In particular, we found the circumstance under which the iteration of a point by projections converges strongly and we answer partially the main question that motivated Bruck’s paper [J. Math. Anal. Appl. 88, 319–332 (1982; Zbl 0512.47042)]. We apply this condition to generalize Prager’s theorem for Hadamard manifolds and generalize Sakai’s theorem for a larger class of the sequences with full measure with respect to Bernoulli measure. In particular, we answer to a long-standing open problem concerning the convergence of the successive projection method [A. Aleyner and S. Reich, J. Convex Anal. 16, No. 3–4, 633–640 (2009; Zbl 1194.47061)]. Furthermore, we study the method of alternating projections for a nested decreasing sequence of convex sets on Hadamard manifolds, and we obtain an alternative proof of the convergence of the proximal point method.
MSC:
| 47Jxx | Equations and inequalities involving nonlinear operators |
| 28D05 | Measure-preserving transformations |
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
Bernoulli measure; Hadamard space; convex feasibility problem; alternating projections; quasi-normal sequence; strong convergenceReferences:
| [1] | Aleyner, A.; Reich, S., Random products of quasi-nonexpansive mappings in Hilbert space, J. Convex Anal., 16, 633-640 (2009) · Zbl 1194.47061 |
| [2] | Amemiya, I.; Ando, T., Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged), 26, 239-244 (1965) · Zbl 0143.16202 |
| [3] | Ariza-Ruiz, D.; Fernández-León, G.; López-Acedo, G.; Nicolae, A., Chebyshev sets in geodesic spaces, J. Approx. Theory, 207, 265-282 (2016) · Zbl 1347.41037 · doi:10.1016/j.jat.2016.02.019 |
| [4] | Ariza-Ruiz, D.; López-Acedo, G.; Nicolae, A., The asymptotic behavior of the composition of firmly nonexpansive mappings, J. Optim. Theory Appl., 167, 2, 409-429 (2015) · Zbl 1329.47052 · doi:10.1007/s10957-015-0710-3 |
| [5] | Bačák, M., Computing medians and means in Hadamard spaces, SIAM J. Optim., 24, 3, 1542-1566 (2014) · Zbl 1306.49046 · doi:10.1137/140953393 |
| [6] | Bačák, M.; Reich, S., The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces, J. Fixed Point Theory, 16, 189-202 (2014) · Zbl 1339.47074 · doi:10.1007/s11784-014-0202-3 |
| [7] | Bačák, M.; Searston, I.; Sims, B., Alternating projections in \(CAT(0)\) spaces, J. Math. Anal. Appl., 385, 2, 599-607 (2012) · Zbl 1232.47047 |
| [8] | Banert, S., Backward-backward splitting in Hadamard spaces, J. Math. Anal. Appl., 414, 656-665 (2014) · Zbl 1307.47076 · doi:10.1016/j.jmaa.2014.01.054 |
| [9] | Bauschke, HH; Borwein, JM, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38, 367-426 (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710 |
| [10] | Bauschke, HH; Matoušková, E.; Reich, S., Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal. Theory Methods Appl., 56, 5, 715-738 (2004) · Zbl 1059.47060 · doi:10.1016/j.na.2003.10.010 |
| [11] | Bento, GC; Melo, JG, Subgradient method for convex feasibility on Riemannian manifolds, J. Optim. Theory Appl., 152, 3, 773-785 (2012) · Zbl 1270.90101 · doi:10.1007/s10957-011-9921-4 |
| [12] | Bregman, LM, Finding the common point of convex sets by the method of successive projection, Soviet Math. Dokl., 6, 688-692 (1965) · Zbl 0142.16804 |
| [13] | Bridson, MR; Haefliger, A., Metric Spaces of Non-positive Curvature (1999), Berlin: Springer, Berlin · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9 |
| [14] | Bruck, RE, Random products of contractions in metric and Banach spaces, J. Math. Anal. Appl., 88, 319-322 (1982) · Zbl 0512.47042 · doi:10.1016/0022-247X(82)90195-0 |
| [15] | Bruck, RE; Reich, S., Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math., 3, 459-470 (1977) · Zbl 0383.47035 |
| [16] | Censor, Y.; Lent, A., Cyclic subgradient projections, Math. Program., 24, 233-235 (1982) · Zbl 0491.90077 · doi:10.1007/BF01585107 |
| [17] | Censor, Y.; Zenios, SA, Parallel Optimization (1997), New York, NY: Oxford University Press, New York, NY · Zbl 0945.90064 |
| [18] | Combettes, PL; Trussell, HJ, Method of successive projections for finding a common point of sets in metric spaces, J. Optim. Theory Appl., 67, 3, 487-507 (1990) · Zbl 0696.90052 · doi:10.1007/BF00939646 |
| [19] | Ferreira, OP; Oliveira, PR, Proximal point algorithm on Riemannian manifolds, Optimization, 51, 257-270 (2002) · Zbl 1013.49024 · doi:10.1080/02331930290019413 |
| [20] | Halperin, I., The product of projection operators, Acta Sci. Math. (Szeged), 23, 96-99 (1962) · Zbl 0143.16102 |
| [21] | Hundal, HS, An alternating projection that does not converge in norm, Nonlinear Anal. Theory Methods Appl., 57)(1), 35-61 (2004) · Zbl 1070.46013 · doi:10.1016/j.na.2003.11.004 |
| [22] | Jost, J., Equilibrium maps between metric spaces, Calc. Var. Part. Differ. Equ., 2, 2, 173-204 (1994) · Zbl 0798.58021 · doi:10.1007/BF01191341 |
| [23] | Kopecká, E., Spokes, mirrors and alternating projections, Nonlinear Anal. Theory Methods Appl., 68, 6, 1759-1764 (2008) · Zbl 1139.46024 · doi:10.1016/j.na.2007.01.006 |
| [24] | Kopecká, E.; Paszkiewicz, A., Strange products of projections, Israel J. Math., 219, 1, 271-286 (2017) · Zbl 1373.46016 · doi:10.1007/s11856-017-1480-4 |
| [25] | Li, C.; López, G.; Martín-Márquez, V., Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc., 79, 3, 663-683 (2009) · Zbl 1171.58001 · doi:10.1112/jlms/jdn087 |
| [26] | Matoušková, E.; Reich, S., The Hundal example revisited, J. Nonlinear Convex Anal., 4, 3, 411-427 (2003) · Zbl 1065.47048 |
| [27] | Prager, M., On a principle of convergence in a Hilbert space, Czechoslovak Math. J., 10, 1960, 271-282 (1960) · Zbl 0090.08902 · doi:10.21136/CMJ.1960.100409 |
| [28] | Petersen, K., Ergodic Theory (1983), Cambridge: Cambridge University Press, Cambridge · Zbl 0507.28010 · doi:10.1017/CBO9780511608728 |
| [29] | Sakai, M., Strong convergence of infinite products of orthogonal projections in Hilbert space, Appl. Anal., 59, 1-4, 109-120 (1995) · Zbl 0846.41026 · doi:10.1080/00036819508840393 |
| [30] | Suparatulatorn, R.; Cholamjiak, P.; Suantai, S., On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Methods Softw., 32, 1, 182-192 (2017) · Zbl 06734174 · doi:10.1080/10556788.2016.1219908 |
| [31] | Sakai, T., Riemannian Geometry, Translations of Mathematical Monographs 149 (1996), Providence: American Mathematical Society, Providence · Zbl 0886.53002 |
| [32] | Tseng, P., On the convergence of the products of firmly nonexpansive mappings, SIAM J. Optim., 3, 425-434 (1992) · Zbl 0763.49011 · doi:10.1137/0802021 |
| [33] | Tseng, P.; Bertsekas, DP, Relaxation methods for problems with strictly convex separable costs and linear constraints, Math. Program., 38, 303-321 (1987) · Zbl 0636.90072 · doi:10.1007/BF02592017 |
| [34] | Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications, vol. 297. Kluwer Academic Publishers, Netherlands (2013) |
| [35] | von Neumann, J.: Functional Operators II: The Geometry of Orthogonal Spaces, Princeton University Press, Princeton, NJ, (1950) (This is a reprint of mimeographed lecture notes first distributed in 1933.) |
| [36] | Viana, M.; Oliveira, K., Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1369.37001 · doi:10.1017/CBO9781316422601 |
| [37] | Walter, R., On the metric projection onto convex sets in Riemannian spaces, Arch. Math., 25, 91-98 (1974) · Zbl 0311.53054 · doi:10.1007/BF01238646 |
| [38] | Wang, XM; Li, C.; Yao, JC, Subgradient projection algorithms for convex feasibility on Riemannian manifolds with lower bounded curvatures, J. Optim. Theory Appl., 164, 1, 202-217 (2015) · Zbl 1308.90131 · doi:10.1007/s10957-014-0568-9 |
| [39] | Wang, X.; Li, C.; Wang, J.; Yao, JC, Linear convergence of subgradient algorithm for convex feasibility on Riemannian manifolds, SIAM J. Optim., 25, 4, 2334-2358 (2015) · Zbl 1326.65072 · doi:10.1137/14099961X |
| [40] | Youla, DC; Webb, H., Image restoration by the method of convex projections: part 1-theory, IEEE Trans. Med. Imag., 1, 2, 81-94 (1982) · doi:10.1109/TMI.1982.4307555 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
