A Relaxed Projection Method for Split Variational Inequalities

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Abstract

We study the recently introduced split variational inequality under the framework of variational inequalities in a product space. The feature of our equivalent formulation of split variational inequality is its variable separability (that is, splitting nature) together with a linear constraint. We propose a relaxed projection method, which fully exploits the splitting structure of split variational inequality and which is not only easily implementable, but also globally convergent under some mild conditions. Our numerical results on finding the minimum-norm solution of the split feasibility problem and on solving a separable and convex quadratic programming problem verify the efficiency and stability of our new method.

Keywords

Split variational inequality Projection method Monotone operator Split feasibility problem Separable structure 

Mathematics Subject Classification

65K15 49J40 90C25 47J25 
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Notes

Acknowledgments

The authors were grateful to the two anonymous referees for their valuable comments and suggestions which improved the presentation of the paper. The first two authors were supported by NSFC (11171083, 11301123), Zhangjiang Provincial NSFC LZ14A010003 and Research Foundation of Hangzhou Dianzi University (KYS075612037). The third author was supported in part by NSC 102-2115-M-110-001-MY3.

References

  1. 1.
    Fichera, G.: Sul pproblem elastostatico di signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 34, 138–142 (1963)Google Scholar
  2. 2.
    Stampacchia, G.: Formes bilinéaires coercitives sur les ensembles convexes. CR Acad. Sci. Paris 258, 4413–4416 (1964)MATHMathSciNetGoogle Scholar
  3. 3.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)Google Scholar
  4. 4.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithm 59, 301–323 (2012)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Byrne, C.L.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)MATHMathSciNetCrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 7.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algorithm 8, 221–239 (1994)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    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)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    López, G., Martin-Marquez, V., Wang, F.H., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012)CrossRefGoogle Scholar
  10. 10.
    Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367 (2007)MATHMathSciNetGoogle Scholar
  11. 11.
    Xu, H.K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)MATHCrossRefGoogle Scholar
  12. 12.
    Xu, H.K.: terative methods for the split feasibility problem in infinite dimensional Hilbert spaces. nverse Probl. 26, 105018 (2010)CrossRefGoogle Scholar
  13. 13.
    Yang, Q.Z.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)MATHCrossRefGoogle Scholar
  14. 14.
    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, 115001 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhang, W.X., Han, D.R., Yuan, X.M.: An efficient simultaneous method for the constrained multiple-sets split feasibility problem. Comput. Optim. Appl. 52, 825–843 (2012)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Byrne, C.L.: Iterative projection onto convex sets using multiple Bregman distances. Inverse Probl. 15, 1295–1313 (1999)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Censor, Y., Chen, W., Combettes, P.L., Davidi, R., Herman, G.T.: On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput. Optim. Appl. 51, 1065–1088 (2012)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Censor, Y., Davidi, R., Herman, G.T.: Perturbation resilience and superiorization of iterative algorithms. Inverse Probl. 26, 065008 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)MATHMathSciNetGoogle Scholar
  22. 22.
    Censor, Y., Gibali, A., Reich, S.: A von Neumann alternating method for finding common solutions to variational inequalities. Nonlinear Anal. 75, 4596–4603 (2012)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set-Valued Var. Anal. 20, 229–247 (2012)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Colao, V., Marino, G., Xu, H.K.: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64, 81–101 (1994)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Fukushima, M.: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 2, 93–111 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kazmi, K.R., Rizvi, S.H.: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim. Lett. 8, 1113–1124 (2014)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21, 1319–1344 (2011)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Moudafi, A.: Split monotone variational inclusion. J. Optim. Theory Appl. 150, 275–283 (2011)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim. 7, 951–965 (1997)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)MATHGoogle Scholar
  32. 32.
    Kazmi, K.R.: Split nonconvex variational inequality problem. Math. Sci. 7, 36 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kazmi, K.R.: Split general quasi-variational inequality problem. arXiv preprint: arXiv:1308.2750 (2013)
  34. 34.
    He, Z.H.: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012, 1–15 (2012)MATHCrossRefGoogle Scholar
  35. 35.
    Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egypt. Med. Assoc. 21, 44–51 (2013)MATHMathSciNetGoogle Scholar
  36. 36.
    Maingé, P.M.: A viscosity method with no spectral radius requirements for the split common fixed point problem. Eur. J. Oper. Res. 235, 17–27 (2014)MATHCrossRefGoogle Scholar
  37. 37.
    He, B.S.: Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities. Comput. Optim. Appl. 42, 195–212 (2009)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    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)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    He, H.J., Han, D.R., Sun, W.Y., Chen, Y.N.: A hybrid splitting method for variational inequality problems with separable structure. Optim. Method Softw. 28, 725–742 (2013)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Xu, M.H.: Proximal alternating directions method for structured variational inequalities. J. Optim. Theory Appl. 134, 107–117 (2007)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Yuan, X.M.: An improved proximal alternating direction method for monotone variational inequalities with separable structure. Comput. Optim. Appl. 49, 17–29 (2011)MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    He, B.S., Liao, L.Z., Qian, M.J.: Alternating projection based prediction-correction methods for structured variational inequalities. J. Comput. Math. 24, 693–710 (2006)MATHMathSciNetGoogle Scholar
  43. 43.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation, Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)MATHGoogle Scholar
  44. 44.
    Palta, J.R., Mackie, T.R.: Intensity Modulated Radiation Therapy: The State of the Art. Medical Physical Monograph. American Association of Physists in Medicine, vol. 29. Medical Physical Publishing, Madison (2003)Google Scholar
  45. 45.
    Wu, Q., Mohan, R., Niemierko, A., Schmidt-Ullrich, R.: Optimization of intensity-modulated radiotherapy plan based on the equivalent uniform dose. Int. J. Radiat. Oncol. Biol. Phys. 52, 224–235 (2003)CrossRefGoogle Scholar
  46. 46.
    Tao, M., Yuan, X.M.: An inexact parallel splitting augmented Lagrangian method for monotone variational inequalities with separable structures. Comput. Optim. Appl. 52, 439–461 (2012)MATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Ceng, L.C., Ansari, Q.H., Yao, J.C.: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75, 2116–2125 (2012)Google Scholar
  48. 48.
    Han, D.R., He, H.J., Yang, H., Yuan, X.M.: A customized Douglas–Rachford splitting algorithm for separable convex minimization with linear constraints. Numer. Math. 127, 167–200 (2014)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Golub, G.H., von Matt, U.: Quadratically constrained least squares and quadratic problems. Numer. Math. 59, 561–580 (1991)MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Han, D.R.: Inexact operator splitting methods with self-adaptive strategy for variational inequality problems. J. Optim. Theory Appl. 132, 227–243 (2007)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceHangzhou Dianzi UniversityHangzhouChina
  2. 2.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan

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