Proximal point methods for solving mixed variational inequalities on the Hadamard manifolds.

*(English)*Zbl 1244.49019Summary: We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.

##### MSC:

49J40 | Variational inequalities |

PDF
BibTeX
XML
Cite

\textit{M. A. Noor} and \textit{K. I. Noor}, J. Appl. Math. 2012, Article ID 657278, 8 p. (2012; Zbl 1244.49019)

Full Text:
DOI

**OpenURL**

##### References:

[1] | D. Azagra, J. Ferrera, and F. López-Mesas, “Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds,” Journal of Functional Analysis, vol. 220, no. 2, pp. 304-361, 2005. · Zbl 1067.49010 |

[2] | V. Colao, G. Lopez, G. Marino, and V. Martin-Marquez, “Equilibrium problems in Hadamard manifolds,” Journal of Mathematical Analysis and Applications, vol. 388, pp. 61-77, 2012. · Zbl 1273.49015 |

[3] | M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser, Boston, Mass, USA, 1992. |

[4] | O. P. Ferreira and P. R. Oliveira, “Proximal point algorithm on Riemannian manifolds,” Optimization, vol. 51, no. 2, pp. 257-270, 2002. · Zbl 1013.49024 |

[5] | T. Sakai, Riemannian Geometry, vol. 149, American Mathematical Society, Providence, RI, USA, 1996. · Zbl 0886.53002 |

[6] | G. Tang, L. W. Zhou, and N. J. Huang, “The proximal point algorithm for pseudomontone variational inequalities on Hadamard manifolds,” Optimization Letters. In press. |

[7] | C. Udri\cste, Convex Functions and Optimization Methods on Riemannian Manifolds, vol. 297, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. · Zbl 0932.53003 |

[8] | S. Z. Németh, “Variational inequalities on Hadamard manifolds,” Nonlinear Analysis. Theory, Methods & Applications, vol. 52, no. 5, pp. 1491-1498, 2003. · Zbl 1016.49012 |

[9] | R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1981. |

[10] | M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119-122, 1988. · Zbl 0655.49005 |

[11] | M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217-229, 2000. · Zbl 0964.49007 |

[12] | M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199-277, 2004. · Zbl 1134.49304 |

[13] | M. A. Noor, “Fundamentals of mixed quasi variational inequalities,” International Journal of Pure and Applied Mathematics, vol. 15, no. 2, pp. 137-258, 2004. · Zbl 1059.49018 |

[14] | M. A. Noor, “Fundamentals of equilibrium problems,” Mathematical Inequalities & Applications, vol. 9, no. 3, pp. 529-566, 2006. · Zbl 1099.91072 |

[15] | M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182-186, 2009. · Zbl 1163.49303 |

[16] | M. A. Noor, “On an implicit method for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 147, no. 2, pp. 411-417, 2010. · Zbl 1202.90253 |

[17] | M. A. Noor, “Auxiliary principle technique for solving general mixed variational inequalities,” Journal of Advanced Mathematical Studies, vol. 3, no. 2, pp. 89-96, 2010. · Zbl 1222.49013 |

[18] | M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012. · Zbl 1242.49017 |

[19] | M. A. Noor, K. I. Noor, and T. M. Rassias, “Some aspects of variational inequalities,” Journal of Computational and Applied Mathematics, vol. 47, no. 3, pp. 285-312, 1993. · Zbl 0788.65074 |

[20] | M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65-69, 2012. · Zbl 1244.65084 |

[21] | Y. Yao, Y. C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press. · Zbl 1260.47085 |

[22] | Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 6, no. 4, pp. 621-628, 2012. · Zbl 1280.90097 |

[23] | Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. · Zbl 1232.49011 |

[24] | Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, pp. 1506-1515, 2012. · Zbl 1275.47130 |

[25] | Y. Yao, M. A. Noor, Y. C. Liou, and S. M. Kang, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768272, 10 pages, 2012. · Zbl 1232.49012 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.