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Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem

Abstract

Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms to solve these problems have been proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain assumptions.

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References

  1. Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, pp. 473–504. Elsevier, Amsterdam (2001)

    Google Scholar 

  2. Combettes, P.L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process. 51(7), 1771–1782 (2003)

    Article  MathSciNet  Google Scholar 

  3. Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  4. Slavakis, K., Yamada, I.: Robust wideband beamforming by the hybrid steepest descent method. IEEE Trans. Signal Process. 55, 4511–4522 (2007)

    Article  MathSciNet  Google Scholar 

  5. Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. (2010, to appear). doi:10.1007/s10107-010-0427-x

  6. Maingé, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 3, 529–538 (2007)

    MATH  MathSciNet  Google Scholar 

  7. Moudafi, A.: Krasnoselski–Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635–1640 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  8. Zeidler, E.: Nonlinear Functional Analysis ans Its Applications II/B. Nonlinear Monotone Operators. Springer, New York (1985)

    Google Scholar 

  9. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)

    MATH  Article  MathSciNet  Google Scholar 

  10. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003)

    Google Scholar 

  11. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  12. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)

    MATH  Book  Google Scholar 

  13. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, New York (1993)

    Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  15. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics Appl. Math., vol. 28. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  16. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Classics Appl. Math., vol. 31. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  17. Vasin, V.V., Ageev, A.L.: Ill-Posed Problems with A Priori Information. V.S.P. Intl Science, Utrecht (1995)

    MATH  Google Scholar 

  18. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984)

    MATH  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  20. Stark, H., Yang, Y.: Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. Wiley, New York (1998)

    MATH  Google Scholar 

  21. Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J. Optim. 15, 555–572 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  22. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996)

    Google Scholar 

  23. Izmaelov, A.F., Solodov, M.V.: An active set Newton method for mathematical program with complementary constraints. SIAM J. Optim. 19, 1003–1027 (2008)

    Article  MathSciNet  Google Scholar 

  24. Hirstoaga, S.A.: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324, 1020–1035 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  25. Iiduka, H.: Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 71, 1292–1297 (2009)

    Article  MathSciNet  Google Scholar 

  26. Iiduka, H.: A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 59, 873–885 (2010)

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Hideaki Iiduka.

Additional information

Communicated by R. Glowinski.

I am sincerely grateful to Professor Angelo Miele of the Rice University for helping me improve the manuscript. This work was supported by the Japan Society for the Promotion of Science through a Grant-in-Aid (19001979).

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Iiduka, H. Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem. J Optim Theory Appl 148, 580–592 (2011). https://doi.org/10.1007/s10957-010-9769-z

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  • DOI: https://doi.org/10.1007/s10957-010-9769-z

Keywords

  • Hierarchical constrained optimization problem
  • Variational inequality
  • Monotone operator
  • Nonexpansive mapping
  • Fixed point
  • Strong convergence