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On the Minimal Displacement Vector of Compositions and Convex Combinations of Nonexpansive Mappings

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Abstract

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. It was shown in 2012 that if finitely many firmly nonexpansive mappings have or “almost have” fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis–Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results.

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Notes

  1. We shall write \({\text {dom}}A = \big \{{x\in X}~\big |~{Ax\ne \varnothing }\big \}\) for the domain of A, \({\text {ran}}A = A(X) = \bigcup _{x\in X}Ax\) for the range of A, and \({\text {gra}}A=\big \{{(x,u)\in X\times X}~\big |~{u\in Ax}\big \}\) for the graph of A.

  2. Here and elsewhere, \({\text {Id}}\) denotes the identity operator on X.

  3. Given a nonempty closed convex subset C of X, we denote its projection mapping or projector by \({{\text {P}}}_ {C}\).

References

  1. J.B. Baillon, R.E. Bruck, and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston Journal of Mathematics 4 (1978), 1–9.

  2. H.H. Bauschke, The composition of finitely many projections onto closed convex sets in Hilbert space is asymptotically regular, Proceedings of the American Mathematical Society 131 (2003), 141–146.

    Article  MathSciNet  Google Scholar 

  3. H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Second Edition, Springer, 2017.

    Book  Google Scholar 

  4. H.H. Bauschke, W.L. Hare, and W.M. Moursi, Generalized solutions for the sum of two maximally monotone operators, SIAM Journal on Control and Optimization 52 (2014), 1034–1047.

    Article  MathSciNet  Google Scholar 

  5. H.H. Bauschke, V. Martin-Marquez, S.M. Moffat, and X. Wang, Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular, Fixed Point Theory and Applications (2012), 2012:53.

    Article  MathSciNet  Google Scholar 

  6. H.H. Bauschke and W.M. Moursi, The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings, Optimization Letters 12 (2018), 1465–1474.

    Article  MathSciNet  Google Scholar 

  7. H.H. Bauschke and W.M. Moursi, The Douglas–Rachford algorithm for two (not necessarily intersecting) affine subspaces, SIAM Journal on Optimization 26 (2016), 968–985.

    Article  MathSciNet  Google Scholar 

  8. H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, 1973.

  9. H. Brezis and A. Haraux, Image d’une Somme d’opérateurs Monotones et Applications, Israel Journal of Mathematics 23 (1976), 165–186.

    Article  MathSciNet  Google Scholar 

  10. R.S. Burachik and A.N. Iusem, Set-Valued Mappings and Enlargements of Monotone Operators, Springer, 2008.

    MATH  Google Scholar 

  11. P.L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization 53 (2004), 475–504.

    Article  MathSciNet  Google Scholar 

  12. P.L. Combettes and I. Yamada, Compositions and convex combinations of averaged nonexpansive operators, Journal of Mathematical Analysis and Applications 425 (2014), 55–70.

    Article  MathSciNet  Google Scholar 

  13. A. De Pierro, From parallel to sequential projection methods and vice versa in convex feasibility: results and conjectures. In: Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000), 187–201, Studies in Computational Mathematics 8 (2001), North-Holland, Amsterdam.

  14. J. Eckstein and D.P. Bertsekas, On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming 55 (1992), 293–318.

    Article  MathSciNet  Google Scholar 

  15. P. Giselsson, Tight global linear convergence rate bounds for Douglas–Rachford splitting, Journal of Fixed Point Theory and Applications 19 (2017), 2241–2270.

    Article  MathSciNet  Google Scholar 

  16. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.

    Book  Google Scholar 

  17. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, 1984.

  18. U. Kohlenbach, A polynomial rate of asymptotic regularity for compositions of projections in Hilbert space, Foundations of Computational Mathematics 19 (2019), 83–99.

    Article  MathSciNet  Google Scholar 

  19. U. Kohlenbach, G. López-Acedo and A. Nicolae, Quantitative asymptotic regularity results for the composition of two mappings, Optimization 66 (2017), 1291–1299.

    Article  MathSciNet  Google Scholar 

  20. G.J. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Mathematical Journal 29 (1962), 341–346.

    Article  MathSciNet  Google Scholar 

  21. W.M. Moursi, The forward-backward algorithm and the normal problem, Journal of Optimization Theory and Applications 176 (2018), 605–624.

    Article  MathSciNet  Google Scholar 

  22. W.M. Moursi and L. Vandenberghe, Douglas–Rachford splitting for the sum of a Lipschitz continuous and a strongly monotone operator, Journal of Optimization Theory and Applications 183 (2019), 179–198.

    Article  MathSciNet  Google Scholar 

  23. S. Reich, Extension problems for accretive sets in Banach spaces, Journal of Functional Analysis 26 (1977), 378–395.

    Article  MathSciNet  Google Scholar 

  24. S. Reich, The fixed point property for non-expansive mappings, American Mathematical Monthly 83 (1976), 266–268.

    Article  MathSciNet  Google Scholar 

  25. S. Reich, The fixed point property for non-expansive mappings, II, American Mathematical Monthly 87 (1980), 292–294.

    Article  Google Scholar 

  26. S. Reich, On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, Journal of Mathematical Analysis and Applications 79 (1981), 113–126.

    Article  MathSciNet  Google Scholar 

  27. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.

    Book  Google Scholar 

  28. R.T. Rockafellar and R.J-B. Wets, Variational Analysis, Springer-Verlag, corrected 3rd printing, 2009.

  29. S. Simons, Minimax and Monotonicity, Springer, 1998.

    Book  Google Scholar 

  30. S. Simons, From Hahn-Banach to Monotonicity, Springer, 2008.

    MATH  Google Scholar 

  31. C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002.

    Book  Google Scholar 

  32. E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer, 1993.

    MATH  Google Scholar 

  33. E. Zeidler, Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators, Springer, 1990.

    Book  Google Scholar 

  34. E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators, Springer-Verlag, 1990.

    Book  Google Scholar 

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Acknowledgements

The authors thank two anonymous referees for constructive and insightful comments. The research of HHB was partially supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada. The research of WMM was partially supported by a Natural Sciences and Engineering Research Council of Canada Postdoctoral Fellowship.

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Authors and Affiliations

  1. Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Heinz H. Bauschke

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Walaa M. Moursi

Authors
  1. Heinz H. Bauschke
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  2. Walaa M. Moursi
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Correspondence to Heinz H. Bauschke.

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Communicated by Michael Overton.

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Bauschke, H.H., Moursi, W.M. On the Minimal Displacement Vector of Compositions and Convex Combinations of Nonexpansive Mappings . Found Comput Math 20, 1653–1666 (2020). https://doi.org/10.1007/s10208-020-09449-w

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  • DOI: https://doi.org/10.1007/s10208-020-09449-w

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