##
**Lower semicontinuity of the solution map to a parametric vector variational inequality.**
*(English)*
Zbl 1187.90300

Summary: This paper is concerned with the study of solution stability of a parametric vector variational inequality, where mappings may not be strongly monotone. Under some requirements that the operator of a unperturbed problem is monotone or it satisfies degree conditions, we show that the solution map of a parametric vector variational inequality is lower semicontinuous.

### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C31 | Sensitivity, stability, parametric optimization |

Full Text:
DOI

### References:

[1] | Bessis D.N., Ledyaev Yu.S., Vinter R.B.: Dualization of the Euler and Hamiltonian inclusions. Nonlinear Anal. 43, 861–882 (2001) · Zbl 1004.49016 |

[2] | Chen G.Y.: Existence of solutions for a vector variational inequality: an extension of the Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992) · Zbl 0795.49010 |

[3] | Chen G.Y., Craven B.D.: A vector variational inequality and optimization over an efficient set. ZOR-Meth. Models Oper. Res. 34, 1–12 (1990) · Zbl 0693.90091 |

[4] | Deimling K.: Nonlinear functional analysis. Springer-Verlag, Berlin (1985) · Zbl 0559.47040 |

[5] | Fonseca I., Gangbo W.: Degree theory in analysis and applications. Oxford, New York (1995) · Zbl 0852.47030 |

[6] | Giannessi, F.: Theorems of alternative, quadractic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.-L. (eds.) Variational inequality and complementarity problems, Proceedings of an International School of Mathematics on Variational Inequalities and Complementarity Problems in Mathematical Physics and Economics, Erice, 19–30 June 1978 (1978) |

[7] | Ioffe A.D., Tihomirov V.M.: Theory of extremal problems. North-Holland, Amsterdam (1979) · Zbl 0407.90051 |

[8] | Jeyakumar V., Oettli W., Natividad M.: A solvability theorem for a class of Quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993) · Zbl 0791.46002 |

[9] | Kien B.T.: On the metric projection onto a family of closed convex sets in a uniformly convex Banach space. Nonlinear Anal. Forum 7, 93–102 (2002) · Zbl 1106.41301 |

[10] | Kien B.T., Wong M.-M.: On the solution stability of variational inequality. J. Glob. Optim. 39, 101–111 (2007) · Zbl 1160.47052 |

[11] | Kien B.T., Wong N.-C., Yao J.-C.: Generalized vector variational inequalites with star-monotone and discontinuous operators. Nonlinear Anal. 9, 2859–2871 (2008) · Zbl 1336.49010 |

[12] | LLoyd N.G.: Degree theory. Cambridge University Press, Cambridge (1978) · Zbl 0367.47001 |

[13] | Lee G.M., Kim D.S., Lee B.S.: Generalized vector variational inequality. Appl. Math. Lett. 9, 39–42 (1996) · Zbl 0862.49014 |

[14] | Lee G.M., Kim D.S., Lee B.S., Yen N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745–765 (1998) · Zbl 0956.49007 |

[15] | Mangasarian O.L., Shiau T.-H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J. Control Optim. 25, 583–595 (1998) · Zbl 0613.90066 |

[16] | Yang X.Q.: Vector variational inequality and its duality. Nonlinear Anal. 21, 869–877 (1993) · Zbl 0809.49009 |

[17] | Yen N.D.: Hölder continuity of solution to a parametric variational inequality. Appl. Math. Optim 31, 245–255 (1995) · Zbl 0821.49011 |

[18] | Zeidler E.: Nonlinear functional analysis and its application, I: fixed-point theorems. Springer-Verlag, Berlin (1993) · Zbl 0794.47033 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.