163
Views
6
CrossRef citations to date
0
Altmetric
Original Articles

Relaxed hybrid steepest-descent methods with variable parameters for triple-hierarchical variational inequalities

, &
Pages 1793-1810 | Received 07 Jun 2011, Accepted 04 Aug 2011, Published online: 14 Sep 2011

References

  • Baiocchi , C and Capelo , A . 1984 . Variational and Quasivariational Inequalities: Applications to Free Boundary Problems , New York : John Wiley and Sons .
  • Ekeland , I and Temam , R . 1999 . “ Convex Analysis and Variational problems ” . In Classical Applications of Mathematics , Vol. 28 , Philadelphia : SIAM .
  • Facchinei , F and Pang , JS . 2003 . Finite-Dimensional Variational Inequalities and Complementarity Problems I , New York : Springer-Verlag .
  • Kikuchi , N and Oden , JT . 1988 . Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , Philadelphia : SIAM .
  • Kinderlehrer , D and Stampacchia , G . 1980 . An Introduction to Variational Inequalities and their Applications , New York : Academic Press .
  • Konnov , IV . 2007 . Equilibrium Models and Variational Inequalities , Amsterdam : Elsevier B. V. .
  • Patriksson , M . 1999 . Nonlinear Programming and Variational Inequalities: A unified Approach , Dordrecht, Boston, London : Kluwer Academic Publishers .
  • Takahashi , W . 2000 . Nonlinear Functional Analysis , Yokohama : Yokohama Publishers .
  • Glowinski , R , Lions , J-L and Trémolières , R . 1981 . Numerical Analysis and Variational Inequalities , Amsterdam : North-Holland .
  • Konnov , IV . 2001 . Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical Systems , Vol. 495 , Berlin, Heidelberg, New York : Springer-Verlag .
  • Combettes , PL . 2003 . A block-iterative surrogate constraint splitting method for quadratic signal recovery . IEEE Trans. Signal Process , 51 ( 7 ) : 1771 – 1782 .
  • Slavakis , K and Yamada , I . 2007 . Robust wideband beamforming by the hybrid steepest descent method . IEEE Trans. Signal Process , 55 : 4511 – 4522 .
  • Iiduka , H . 2010 . Fixed point optimization algorithm and its application to power control in CDMA data networks . Math. Program. , to appear, doi:10.1007/s10107-010-0427-x
  • Iiduka , H and Yamada , I . 2009 . 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 .
  • Yamada , I . 2001 . “ The hybrid steepest-descent method for the variational inequality problem over the intersection of fixed-point sets of nonexpansive mappings ” . In Inherently Parallel Algorithms in Feasibility and Optimization and their Applications , Edited by: Butnariu , D , Censor , Y and Reich , S . 473 – 504 . Dordrecht , , Netherlands : Kluwer Academic Publishers .
  • Xu , HK and Kim , TH . 2003 . Convergence of hybrid steepest-descent methods for variational inequalities . J. Optim. Theory Appl. , 119 : 185 – 201 .
  • Zeng , LC , Wong , NC and Yao , JC . 2007 . Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities . J. Optim. Theory Appl. , 132 : 51 – 69 .
  • Cabot , A . 2005 . Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization . SIAM J. Optim. , 15 : 555 – 572 .
  • Hirstoaga , SA . 2006 . Iterative selection methods for common fixed point problems . J. Math. Anal. Appl. , 324 : 1020 – 1035 .
  • Iiduka , H . 2010 . A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping . Optimization , 59 : 873 – 885 .
  • Izmaelov , AF and Solodov , MV . 2008 . An active set Newton method for mathematical program with complementarity constraints . SIAM J. Optim. , 19 : 1003 – 1027 .
  • Luo , ZQ , Pang , JS and Ralph , D . 1996 . Mathematical Programs with Equilibrium Constraints , New York : Cambridge University Press .
  • Mainge , PE and Moudafi , A . 2007 . Strong convergence of an iterative method for hierarchical fixed-point problems . Pacific J. Optim. , 3 : 529 – 538 .
  • Moudafi , A . 2007 . Krasnoselski-Mann iteration for hierarchical fixed-point problems . Inverse Probl. , 23 : 1635 – 1640 .
  • Iiduka , H . 2009 . Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem . Nonlinear Anal. , 71 : 1292 – 1297 .
  • Iiduka , H . 2011 . Iterative algorithm for solving Triple-hierarchical constrained optimization problem . J. Optim. Theory Appl. , 148 : 580 – 592 .
  • Ceng , LC , Ansari , QH and Yao , JC . 2011 . Iterative methods for triple hierarchical variational inequalities in Hilbert spaces . J. Optim. Theory Appl. , 151 ( 3 )
  • Zeidler , E . 1985 . Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators , New York : Springer-Verlag .
  • Hiriart-Urruty , JB and Lemarechal , C . 1993 . Convex Analysis and Minimization Algorithms I , New York : Springer-Verlag .
  • Goebel , K and Reich , S . 1984 . Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , New York : Dekker .
  • Vasin , VV and Ageev , AL . 1995 . Ill-Posed Problems with A Priori Information , Utrecht : V.S.P. International Science .
  • Xu , HK . 2002 . Iterative algorithms for nonlinear operators . J. London Math. Soc. , 66 : 240 – 256 .
  • Goebel , K and Kirk , WA . 1990 . Topics on Metric Fixed-Point Theory , Cambridge : Cambridge Studies in Advanced Mathematics, Cambridge University Press .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.