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July  2014, 1(3): 331-346. doi: 10.3934/jdg.2014.1.331

Asymptotic behavior of compositions of under-relaxed nonexpansive operators

Jean-Bernard Baillon 1, , Patrick L. Combettes 2, and  Roberto Cominetti 3,

1. 

Université Paris 1 Panthéon-Sorbonne, SAMM – EA 4543, 75013 Paris, France
2. 

Sorbonne Universités – UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
3. 

Universidad de Chile, Departamento de Ingeniería Industrial, Santiago, Chile

Received  April 2013 Revised  October 2013 Published  July 2014

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
Keywords: fixed point, under-relaxed cycles., projection operator, nonexpansive operator, Cyclic projections, De Pierro's conjecture.
Mathematics Subject Classification: 47H09, 47H10, 47N10, 65K1.
Citation: Jean-Bernard Baillon, Patrick L. Combettes, Roberto Cominetti. Asymptotic behavior of compositions of under-relaxed nonexpansive operators. Journal of Dynamics & Games, 2014, 1 (3) : 331-346. doi: 10.3934/jdg.2014.1.331
References:
[1]

H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270. doi: 10.1137/090754297.  Google Scholar

[2]

J.-B. Baillon, P. L. Combettes and R. Cominetti, There is no variational characterization of the cycles in the method of periodic projections, J. Funct. Anal., 262 (2012), 400-408. doi: 10.1016/j.jfa.2011.09.002.  Google Scholar

[3]

H. H. Bauschke and J. M. Borwein, On the convergence of von Neumann's alternating projection algorithm for two sets, Set-Valued Anal., 1 (1993), 185-212. doi: 10.1007/BF01027691.  Google Scholar

[4]

H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke and H. Wolkowicz, eds., Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-9569-8.  Google Scholar

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[6]

H. H. Bauschke and M. R. Edwards, A conjecture by De Pierro is true for translates of regular subspaces, J. Nonlinear Convex Anal., 6 (2005), 93-116.  Google Scholar

[7]

H. H. Bauschke, X. Wang and C. J. S. Wylie, Fixed points of averages of resolvents: Geometry and algorithms, SIAM J. Optim., 22 (2012), 24-40. doi: 10.1137/110823778.  Google Scholar

[8]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, New York, 1973.  Google Scholar

[9]

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), 15-26. doi: 10.1016/0022-1236(75)90027-0.  Google Scholar

[10]

C. L. Byrne, Applied Iterative Methods, A. K. Peters, Wellesley, MA, 2008.  Google Scholar

[11]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012.  Google Scholar

[12]

Y. Censor, P. P. B. Eggermont and D. Gordon, Strong under-relaxation in Kaczmarz's method for inconsistent systems, Numer. Math., 41 (1983), 83-92. doi: 10.1007/BF01396307.  Google Scholar

[13]

P. L. Combettes, Inconsistent signal feasibility problems: Least-squares solutions in a product space, IEEE Trans. Signal Process., 42 (1994), 2955-2966. doi: 10.1109/78.330356.  Google Scholar

[14]

A. R. De Pierro, From parallel to sequential projection methods and vice versa in convex feasibility: Results and conjectures, in Inherently Parallel Algorithms for Feasibility and Optimization, Elsevier, New York, 2001, 187-201. doi: 10.1016/S1570-579X(01)80012-4.  Google Scholar

[15]

A. R. De Pierro and A. N. Iusem, A parallel projection method for finding a common point of a family of convex sets, Pesquisa Operacional, 5 (1985), 1-20. Google Scholar

[16]

L. G. Gubin, B. T. Polyak and E. V. Raik, The method of projections for finding the common point of convex sets, Comput. Math. Math. Phys., 7 (1967), 1-24. doi: 10.1016/0041-5553(67)90113-9.  Google Scholar

[17]

W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14 (1966), 276-284. doi: 10.1016/0022-247X(66)90027-8.  Google Scholar

[18]

X. Wang and H. H. Bauschke, Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne, Nonlinear Anal., 74 (2011), 4550-4572. doi: 10.1016/j.na.2011.04.024.  Google Scholar

[19]

E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

show all references

References:
[1]

H. Attouch, L. M. Briceño-Arias and P. L. Combettes, A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim., 48 (2010), 3246-3270. doi: 10.1137/090754297.  Google Scholar

[2]

J.-B. Baillon, P. L. Combettes and R. Cominetti, There is no variational characterization of the cycles in the method of periodic projections, J. Funct. Anal., 262 (2012), 400-408. doi: 10.1016/j.jfa.2011.09.002.  Google Scholar

[3]

H. H. Bauschke and J. M. Borwein, On the convergence of von Neumann's alternating projection algorithm for two sets, Set-Valued Anal., 1 (1993), 185-212. doi: 10.1007/BF01027691.  Google Scholar

[4]

H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke and H. Wolkowicz, eds., Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-9569-8.  Google Scholar

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[6]

H. H. Bauschke and M. R. Edwards, A conjecture by De Pierro is true for translates of regular subspaces, J. Nonlinear Convex Anal., 6 (2005), 93-116.  Google Scholar

[7]

H. H. Bauschke, X. Wang and C. J. S. Wylie, Fixed points of averages of resolvents: Geometry and algorithms, SIAM J. Optim., 22 (2012), 24-40. doi: 10.1137/110823778.  Google Scholar

[8]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, New York, 1973.  Google Scholar

[9]

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), 15-26. doi: 10.1016/0022-1236(75)90027-0.  Google Scholar

[10]

C. L. Byrne, Applied Iterative Methods, A. K. Peters, Wellesley, MA, 2008.  Google Scholar

[11]

A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012.  Google Scholar

[12]

Y. Censor, P. P. B. Eggermont and D. Gordon, Strong under-relaxation in Kaczmarz's method for inconsistent systems, Numer. Math., 41 (1983), 83-92. doi: 10.1007/BF01396307.  Google Scholar

[13]

P. L. Combettes, Inconsistent signal feasibility problems: Least-squares solutions in a product space, IEEE Trans. Signal Process., 42 (1994), 2955-2966. doi: 10.1109/78.330356.  Google Scholar

[14]

A. R. De Pierro, From parallel to sequential projection methods and vice versa in convex feasibility: Results and conjectures, in Inherently Parallel Algorithms for Feasibility and Optimization, Elsevier, New York, 2001, 187-201. doi: 10.1016/S1570-579X(01)80012-4.  Google Scholar

[15]

A. R. De Pierro and A. N. Iusem, A parallel projection method for finding a common point of a family of convex sets, Pesquisa Operacional, 5 (1985), 1-20. Google Scholar

[16]

L. G. Gubin, B. T. Polyak and E. V. Raik, The method of projections for finding the common point of convex sets, Comput. Math. Math. Phys., 7 (1967), 1-24. doi: 10.1016/0041-5553(67)90113-9.  Google Scholar

[17]

W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14 (1966), 276-284. doi: 10.1016/0022-247X(66)90027-8.  Google Scholar

[18]

X. Wang and H. H. Bauschke, Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne, Nonlinear Anal., 74 (2011), 4550-4572. doi: 10.1016/j.na.2011.04.024.  Google Scholar

[19]

E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

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