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A new iteration technique for nonlinear operators as concerns convex programming and feasibility problems

  • D. R. Sahu
  • A. PiteaEmail author
  • M. Verma
Original Paper
  • 30 Downloads

Abstract

The aim of this work is to develop an S-iteration technique for finding common fixed points for nonself quasi-nonexpansive mappings in the framework of a uniformly convex Banach space. Convergence properties of the proposed algorithm are analyzed in the setting of uniformly convex Banach spaces. To prove the usability of our results, some novel applications are provided, focused on zeros of accretive operators, convex programming, and feasibility problems. Some numerical experiments with real datasets for Lasso problems are provided.

Keywords

Accretive operator Convex programming Feasibility problem Fixed point S-iteration process 

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Notes

Acknowledgements

The second author has been funded by University Politehnica of Bucharest, through the “Excellence Research Grants” Program, UPB - GEX. Identifier: UPB-EXCELENŢĂ-2017, ID 53, no. int. SA 541702, “Analiză neliniară şi optimizări.”

References

  1. 1.
    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed point theory for Lipschitzian-type mappings with applications. Springer, New York (2009)zbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61–79 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities, Application to Free Boundary Problems. Wiley, New York (1984)zbMATHGoogle Scholar
  4. 4.
    Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L., Reich, S.: The asymptotic behavior of the composition of two resolvents. Nonlinear Anal. 60(2), 283–301 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berinde, V.: Iterative approximation of fixed points. Springer, Berlin (2007)zbMATHGoogle Scholar
  7. 7.
    Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688–692 (1965)zbMATHGoogle Scholar
  8. 8.
    Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland (1973)Google Scholar
  9. 9.
    Browder, F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Amer. Math. Soc. 74, 660–665 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Browder, F.E.: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Amer. Math. Soc. 73, 875–882 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cioranescu, I.: Geometry of Banach spaces, duality mapping and nonlinear problems. Kluwer, Amsterdam (1990)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ceng, L.C., Cubiotti, P., Yao, J.C.: Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal. 67, 1463–1473 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chang, S.S., Wang, G., Wang, L., Tang, Y.K., Ma, Z.L.: △-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces. Appl. Math. Comput. 249, 535–540 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chang, S.S., Wang, L., Joseph Lee, H.W., Chan, C.K.: Strong and △-convergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2013, 122 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chidume, C.E., Takens, F.: Geometric properties of Banach spaces and nonlinear iterations, vol. 1965. Springer, London, UK (2009)Google Scholar
  16. 16.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model Simul. 4, 1168–1200 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cholamjiak, P., Abdou, A.A., Cho, Y.J.: Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2015, 227 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Iiduka, H.: Proximal point algorithms for nonsmooth convex optimization with fixed point constraints. Eur. J. Oper. Res. 253(2), 503–513 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kim, G.E.: Weak and strong convergence theorems of quasi-nonexpansive mappings in a Hilbert space. J. Optim. Theory Appl. 152, 727–738 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4.3, 506–510 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Martinet, B.: Determination approchée d’un point fixe d’une application pseudo-contractante. C. R. Acad. Sci Paris Ser. A-B 274, 163–165 (1972)zbMATHGoogle Scholar
  22. 22.
    Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970)zbMATHGoogle Scholar
  23. 23.
    von Neumann, J.: Functional Operators. II. The Geometry of Orthogonal Spaces. Annals of Math. Studies, no. 22. Princeton University Press, Princeton (1950)Google Scholar
  24. 24.
    Nagurney, A.: Network economics: a variational inequality approach. Kluwer Academic Publishers, Dordrecht (1999)CrossRefzbMATHGoogle Scholar
  25. 25.
    Parikh, N., Boyd, S.: Proximal algorithms. Foundations and Trends in Optimization 1.3, 127–239 (2014)CrossRefGoogle Scholar
  26. 26.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper Res. 1, 97–116 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sahu, D.R., Ansari, Q.H., Yao, J.C.: Convergence of inexact Mann iterations generated by nearly nonexpansive sequences and applications. Numer. Funct. Anal. Optim. 37, 1312–1338 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sahu, D.R., Ansari, Q.H., Yao, J.C.: The prox-Tikhonov-like forward-backward method and applications. Taiwan J. Math. 19, 481–503 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sahu, D.R.: On generalized Ishikawa iteration process and nonexpansive mappings in Banach spaces. Demonstratio Math. 36(3), 721–734 (2003)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Shahzad, N.: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Anal. 61, 1031–1039 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Suparatulatorn, R., Cholamjiak, P.: The modified S-iteration process for nonexpansive mappings in CAT(k) spaces. Fixed Point Theory Appl. 2016, 25 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Takahashi, W.: Nonlinear functional analysis, fixed point theory and its applications. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
  35. 35.
    Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Verma, M., Shukla, K.K.: A new accelerated proximal technique for regression with high-dimensional datasets. Knowl. Inf. Syst. 53, 423–438 (2017)CrossRefGoogle Scholar
  37. 37.
    Xu, H.K.: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 314, 631–643 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yao, Y., Cho, Y.J., Liou, Y.C.: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212(2), 242–250 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zegeye, H., Shahzad, N.: Strong convergence theorems for a common zero of a finite family of m-accretive mappings. Nonlinear Anal. 66, 1161–1169 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhang, Q.N., Song, Y.S.: Halpern type proximal point algorithm of accretive operators. Nonlinear Anal. 75, 1859–1868 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhao, L.C., Chang, S.S., Kim, J.K.: Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory Appl. 2013, 353 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Zhao, X., Sahu, D.R., Wen, C.-F.: Iterative methods for system of variational inclusions involving accretive operators and applications. Fixed Point Theory 19(2), 801–822 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsBanaras Hindu UniversityVaranasiIndia
  2. 2.Department of Mathematics and InformaticsUniversity Politehnica of BucharestBucharestRomania
  3. 3.Institute for Development and Research in Banking Technology (IDRBT)HyderabadIndia

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