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

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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.

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References

  1. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed point theory for Lipschitzian-type mappings with applications. Springer, New York (2009)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  3. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities, Application to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  4. Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. Springer, Berlin (2011)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  6. Berinde, V.: Iterative approximation of fixed points. Springer, Berlin (2007)

    MATH  Google Scholar 

  7. Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688–692 (1965)

    MATH  Google Scholar 

  8. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland (1973)

  9. Browder, F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Amer. Math. Soc. 74, 660–665 (1968)

    MathSciNet  MATH  Google Scholar 

  10. Browder, F.E.: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Amer. Math. Soc. 73, 875–882 (1967)

    MathSciNet  MATH  Google Scholar 

  11. Cioranescu, I.: Geometry of Banach spaces, duality mapping and nonlinear problems. Kluwer, Amsterdam (1990)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Chidume, C.E., Takens, F.: Geometric properties of Banach spaces and nonlinear iterations, vol. 1965. Springer, London, UK (2009)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  18. Iiduka, H.: Proximal point algorithms for nonsmooth convex optimization with fixed point constraints. Eur. J. Oper. Res. 253(2), 503–513 (2016)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  20. Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4.3, 506–510 (1953)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  22. Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970)

    MATH  Google Scholar 

  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. Nagurney, A.: Network economics: a variational inequality approach. Kluwer Academic Publishers, Dordrecht (1999)

    MATH  Google Scholar 

  25. Parikh, N., Boyd, S.: Proximal algorithms. Foundations and Trends in Optimization 1.3, 127–239 (2014)

    Google Scholar 

  26. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    MathSciNet  MATH  Google Scholar 

  27. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper Res. 1, 97–116 (1976)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  31. Sahu, D.R.: On generalized Ishikawa iteration process and nonexpansive mappings in Banach spaces. Demonstratio Math. 36(3), 721–734 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Shahzad, N.: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Anal. 61, 1031–1039 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Suparatulatorn, R., Cholamjiak, P.: The modified S-iteration process for nonexpansive mappings in CAT(k) spaces. Fixed Point Theory Appl. 2016, 25 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Takahashi, W.: Nonlinear functional analysis, fixed point theory and its applications. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  36. Verma, M., Shukla, K.K.: A new accelerated proximal technique for regression with high-dimensional datasets. Knowl. Inf. Syst. 53, 423–438 (2017)

    Google Scholar 

  37. Xu, H.K.: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 314, 631–643 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)

    MathSciNet  MATH  Google Scholar 

  39. Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  42. Zhang, Q.N., Song, Y.S.: Halpern type proximal point algorithm of accretive operators. Nonlinear Anal. 75, 1859–1868 (2012)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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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.”

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Authors and Affiliations

  1. Department of Mathematics, Banaras Hindu University, Varanasi, India

    D. R. Sahu

  2. Department of Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, Romania

    A. Pitea

  3. Institute for Development and Research in Banking Technology (IDRBT), Hyderabad, India

    M. Verma

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  1. D. R. Sahu
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  2. A. Pitea
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  3. M. Verma
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Correspondence to A. Pitea.

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Sahu, D.R., Pitea, A. & Verma, M. A new iteration technique for nonlinear operators as concerns convex programming and feasibility problems. Numer Algor 83, 421–449 (2020). https://doi.org/10.1007/s11075-019-00688-9

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  • DOI: https://doi.org/10.1007/s11075-019-00688-9

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