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Solving proximal split feasibility problems without prior knowledge of operator norms

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Abstract

In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function \(f\) such that its image under a bounded linear operator \(A\) minimizes another convex function \(g\). Based on an idea introduced in Lopez (Inverse Probl 28:085004, 2012), we propose a split proximal algorithm with a way of selecting the step-sizes such that its implementation does not need any prior information about the operator norm. Because the calculation or at least an estimate of the operator norm \(\Vert A\Vert \) is not an easy task. Secondly, we investigate the case where one of the two involved functions is prox-regular, the novelty of this approach is that the associated proximal mapping is not nonexpansive any longer. Such situation is encountered, for instance, in numerical solution to phase retrieval problem in crystallography, astronomy and inverse scattering Luke (SIAM Rev 44:169–224, 2002) and is therefore of great practical interest.

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References

  1. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1–14 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brézis, H.: Analyse fonctionnelle, théorie et applications. Masson, Paris (1983)

    MATH  Google Scholar 

  4. Byrne, Ch.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byrne, C.H., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

  7. Chaux, C., Combettes, P.L., Pesquet, J.-C., Wajs, V.R.: A variational formulation for frame-based inverse problems. Inverse Probl. 23, 1495–1518 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clarke, Z.F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)

    MATH  Google Scholar 

  9. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lopez, G., Martin-Marquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012)

    Article  MathSciNet  Google Scholar 

  11. Luke, D.R., Burke, J.V., Lyon, R.G.: Optical wavefront reconstruction: theory and numerical methods. SIAM Rev. 44, 169–224 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Luke, D.R.: Finding best approximation pairs relative to a convex and prox-regular set in Hilbert space. SIAM J. opt. 19(2), 714–729 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mallat, S.G.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, New York (1999)

    MATH  Google Scholar 

  14. Moudafi, A.: Split Monotone Variational Inclusions. J. Optim. Theory Appl. 150(2), 275–283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Moudafi, A.: A proximal iterative approach to a nonconvex optimization problem. Nonlinear Anal. Theory Methods Appl. 72, 704–709 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  17. Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Am. Math. Soc. 348, 1805–1838 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Polyak, B.T.: Introduction to Optimization. Optimization Software Inc., New York (1987)

    Google Scholar 

  19. Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1988)

    Google Scholar 

  20. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877898 (1976)

    Google Scholar 

  21. Xu, H.K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)

    Article  MATH  Google Scholar 

  22. Xu, H.-K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl. 26(10), article 105018 (2010)

    Google Scholar 

  23. Yao, Y., Chen, R., Marino, G., Liou, Y.C.: Applications of fixed point and optimization methods to the multiple-sets split feasibility problem. J. Appl. Math. 2012 Article ID 927530 (2012)

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Acknowledgments

The authors would like to thank the anonymous referees for their careful reading, suggestions and pertinent questions which permitted us to improve the first version of this paper.

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Authors and Affiliations

  1. Ceregmia-Département scientifique, Université des Antilles et de la Guyane, Schoelcher, 97275, Martinique

    A. Moudafi

  2. School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, 492010, India

    B. S. Thakur

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  1. A. Moudafi
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  2. B. S. Thakur
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Correspondence to A. Moudafi.

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Moudafi, A., Thakur, B.S. Solving proximal split feasibility problems without prior knowledge of operator norms. Optim Lett 8, 2099–2110 (2014). https://doi.org/10.1007/s11590-013-0708-4

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