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An efficient simultaneous method for the constrained multiple-sets split feasibility problem

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Abstract

The multiple-sets split feasibility problem (MSFP) captures various applications arising in many areas. Recently, by introducing a function measuring the proximity to the involved sets, Censor et al. proposed to solve the MSFP via minimizing the proximity function, and they developed a class of simultaneous methods to solve the resulting constrained optimization model numerically. In this paper, by combining the ideas of the proximity function and the operator splitting methods, we propose an efficient simultaneous method for solving the constrained MSFP. The efficiency of the new method is illustrated by some numerical experiments.

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References

  1. Attouch, H., Briceño-Arias, H.L., Combettes, P.L.: A parallel splitting method for coupled monotone inclusions. SIAM J. Control Optim. 48(5), 3246–3270 (2010)

    Article  MATH  Google Scholar 

  2. Baillon, J., Haddad, G.: Quelques propriétés des opérateurs angel-bornés et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Byrne, C., Censor, Y.: Proximity function minimization using multiple Bergman projections, with applications to split feasibility and Kullback-Leibler distance minimization. Ann. Oper. Res. 105, 77–98 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York (1997)

    MATH  Google Scholar 

  10. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–270 (1996)

    Article  Google Scholar 

  13. Eckstein, J., Fukushima, M.: Some reformulation and applications of the alternating directions method of multipliers. In: Hager, W.W., et al. (eds.) Large Scale Optimization: State of the Art, pp. 115–134. Kluwer Academic, Dordrecht (1994)

    Chapter  Google Scholar 

  14. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, Berlin (2003)

    Google Scholar 

  15. Fukushima, M.: Application of the alternating directions method of multipliers to separable convex programming problems. Comput. Optim. Appl. 2, 93–111 (1992)

    Article  MathSciNet  Google Scholar 

  16. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, pp. 299–331. North-Holland, Amsterdam (1983)

    Chapter  Google Scholar 

  17. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 17–40 (1976)

    Article  MATH  Google Scholar 

  18. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

  19. Han, D.R.: Inexact operator splitting methods with self-adaptive strategy for variational inequality problems. J. Optim. Theory Appl. 132, 227–243 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, D.R., Xu, W., Yang, H.: An operator splitting method for variational inequalities with partially unknown mappings. Numer. Math. 111, 207–237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. He, B.S.: Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities. Comput. Optim. Appl. 42, 195–212 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, B.S., Liao, L.Z., Han, D.R., Yang, H.: A new inexact alternating direction method for monotone variational inequalities. Math. Program. 92, 103–118 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. He, B.S., Liao, L.Z., Qian, M.J.: Alternating projection based prediction-correction method for structured variational inequalities. J. Comput. Math. 24, 693–710 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Jiang, Z.K., Yuan, X.M.: New parallel descent-like method for solving a class of variational inequalities. J. Optim. Theory Appl. 145, 311–323 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29–53 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Qu, B., Xiu, N.H.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qu, B., Xiu, N.H.: A new halfspace-relaxation projection method for the split feasibility problem. Linear Algebra Appl. 428, 1218–1229 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: Lecture Notes in Comput. Sci., vol. 5567, pp. 464–476 (2009)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Yang, Q.Z.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)

    Article  MATH  Google Scholar 

  31. Ye, C.H., Yuan, X.M.: A descent method for structured monotone variational inequalities. Optim. Methods Softw. 22, 329–338 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yuan, X.M.: An improved proximal alternating direction method for structured variational inequalities. Comput. Optim. Appl. 49, 17–29 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, W.X., Han, D.R., Li, Z.B.: A self-adaptive projection method for solving the multiple-sets split feasibility problem. Inverse Probl. 25. doi:10.1088/0266-5611/25/11/115001

  34. Zhao, J.L., Yang, Q.Z.: Several solution methods for the split feasibility problem. Inverse Probl. 21, 1791–1799 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Wenxing Zhang

  2. School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, 210046, People’s Republic of China

    Deren Han

  3. Department of Mathematics, Hong Kong Baptist University, Hong Kong, People’s Republic of China

    Xiaoming Yuan

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  1. Wenxing Zhang
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  2. Deren Han
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  3. Xiaoming Yuan
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Correspondence to Xiaoming Yuan.

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Zhang, W., Han, D. & Yuan, X. An efficient simultaneous method for the constrained multiple-sets split feasibility problem. Comput Optim Appl 52, 825–843 (2012). https://doi.org/10.1007/s10589-011-9429-8

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  • DOI: https://doi.org/10.1007/s10589-011-9429-8

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